Kronecker’s divisor theory and the generalization of notions and theorems of classical algebraic number theory to Krull domains

Friedemann Lucius

Georg-August-Universität Göttingen Mathematisches Institut Bunsenstraße 3–5 D-37073 Göttingen

e-mail: flucius@aol.com

Introduction

Introductions to algebraic number theory often regard the classical theory of algebraic number fields just as a special case of the theory of Dedekind domains. The level of abstraction can even be more increased by studying Krull domains instead of Dedekind domains. Rings of this type, also known as rings with divisor theory, still allow some sort of prime factorization, but in general, the corresponding system of prime divisors can no longer be made explicit by the system of prime ideals. It has become standard to describe the prime divisors of a Krull domain by its 𝔭-adic valuations, i. e. by its discrete rank one valuations. Thus, Krull domains are usually studied by means of valuation theory and local-global arguments. This methodical approach, however, has the disadvantage of being more abstract and less graphic than classical ideal theory. Compared to prime ideals discrete rank one valuations are indeed rather abstract algebraic objects. In addition, there is no natural definition of the norm of a divisor or the different of a Krull domain with respect to a finite field extension.

The intention of this paper is to present and propagate a methodical alternative devoid of these inconveniences, namely Kronecker’s divisor theory. There have been several attempts to make Kronecker’s methods popular, but most of the treatments on Kronecker’s divisor theory lack of systematic approach. This may explain why Kronecker’s approach to algebraic number theory is little known and still lives in the shadow of ideal and valuation theory.

In [7] Kronecker’s theory was analyzed in modern terms, i. e. by means of the theory of divisorial ideals, Kronecker function rings and Nagata domains. In the present paper we want to show how Kronecker’s divisor theory can be used to transfer notions and well-known results from the theory of Dedekind domains to Krull domains thereby giving an alternative approach to the fundamentals of algebraic number theory.

If D is a Krull domain with field of fractions K, the corresponding Kronecker function ring D^{v} with respect to the v-operation is a principal ideal domain whose elements deliver a complete and (up to association) uniquely determined system of divisors for D in a most natural way. It is well-known that the group of fractional D^{v}-ideals is isomorphic to the Lorenzen v-group Λ_{v}(D) of D, i. e. the group of fractional divisorial ideals of D. We will show that this isomorphism is given by extending divisorial D-ideals to D^{v}-ideals and by contracting D^{v}-ideals to D-ideals. Observing that the integral closure T of D in a finite field extension L/K is again a Krull domain and that the integral closure of D^{v} in the corresponding extension L(X)/K(X) is given by the Kronecker function ring T^{v}, we define the norm of a divisor of T with respect to L/K in line with the usual norm with respect to L(X)/K(X). Since the Kronecker function rings D^{v} resp. T^{v} are principal ideal domains, hence Dedekind domains, the same translation process can be applied to the definition of the residue class field, the ramification index, the inertial degree, the different, the discriminant etc. Consequently, many results on Dedekind domains such as the different and discriminant theorems can easily be transferred to Krull domains. Finally, we will show that the different of T^{v} over D^{v} is generated by an element different diff(F), F ∈ T^{v}, which implies that T^{v} is a simple extension ring of D^{v} with T^{v} = D^{v}[F]. From this we will explain, why, whith slight modifications, Kummer’s method to factorize rational primes in an algebraic number field is also applicable to common inessential discriminant divisors.

The present paper generalizes several results already obtained by H. Flanders for Dedekind domains (cf. [2]). It is a contribution to an adequate assessment of Kronecker’s method. The following passage quoted from the introduction of Flander’s paper explains why Kronecker’s approach to algebraic number theory deserves being carefully studied even today:

In many situations it is extremely convenient, indeed almost imperative, to have a principal ideal ring instead of a Dedekind ring. The usual modern device for passing to this technically vastly simpler situation is to localize either by passing to p-adic completions or by forming the quotient ring with respect to the complement of a finite set of prime ideals. [Kronecker’s divisor …] theory has not generally been looked upon as a tool for accomplishing this reduction to principal ideals, […]. In a certain sense it accomplishes the task much better than does localization because with localization the bulk of the structure of the ideal group is lost, whereas with forms [i. e. elements of the Kronecker function ring] this structure is preserved down to the finest detail. [2, 92]

1 Divisorial ideals, Kronecker divisors and the norm

We start this section with a short introduction to the theory of divisorial ideals and Kronecker divisors as outlined in [7]. If not otherwise stated, we always mean fractional ideals with respect to an integral domain when speaking of ideals.

Let D be a domain with quotient field K. If 𝔞 is a D-ideal and if 𝔞^{−1} means the ideal quotient (D : 𝔞) := {μ ∈ K : μ𝔞 ⊆ D}, we call the fractional D-ideal 𝔞_{v} := (𝔞^{−1})^{−1} the divisorial or v-ideal corresponding to 𝔞. The map 𝔞 ↦ 𝔞_{v} is referred to as the v-operation with respect to D. It induces the t-operation, which is given by

𝔞 ↦ 𝔞_{t} := ⋂_{\begin{matrix}𝔢 f. g. \\ 𝔢⊆𝔞 \end{matrix}}𝔢_{v},

where the abbreviation “f. g.” stands for “finitely generated D-ideal”. The v-operation is said to be of finite type if it coincides with the t-operation. Note that v- and toperation mean the same when restricted to the set of finitely generated D-ideals. If 𝔞 = 𝔞_{v}, the D-ideal 𝔞 is called divisorial or v-ideal; if 𝔞 = 𝔞_{t}, it is called t-ideal. In the next paragraph the symbol “∗” is used to denote both the v- and the t-operation on D.

A ∗-ideal 𝔞 is ∗-finite if there is a finitely generated D-ideal 𝔟 with 𝔞 = 𝔟_{∗}. If this ideal 𝔟 is generated by a_{1}, …, a_{m} ∈ 𝔟, we also write 𝔞 = 𝔞_{∗} = (a_{1}, …, a_{m})_{∗} and refer to a_{1}, …, a_{m} as the ∗-generating system of 𝔞. The fractional ∗-ideals of D form a partially ordered semigroup by the so-called ∗-multiplication “×”. It is defined by 𝔞_{∗}×𝔟_{∗} := (𝔞′ · 𝔟′)_{∗}, where 𝔞′, 𝔟′ denote fractional D-ideals with 𝔞′_{∗} = 𝔞_{∗} and 𝔟′_{∗} = 𝔟_{∗}. The meaning of ∗-divisibility and ∗-invertibility is obvious. Note that, in general, the v-inverse of a v-invertible v-ideal 𝔞 is not v-finite, even if 𝔞 is v-finite. The t-inverse of a t-invertible t-ideal, however, is always t-finite.

If f(X) = a_{0} + a_{1}X + … + a_{m}X^{m} is a polynomial in K[X], the v-ideal

c(f)_{v} := (a_{1}, …, a_{m})_{v}

is called the v-content of f. If c(f)_{v} = D, f is called v-primitive. The set of vprimitive polynomials is denoted by N_{v}(D). The v-content is said to be multiplicative if c(f · g)_{v} = c(f)_{v}×c(g)_{v} for all f, g ∈ K[X]. The v-content is multiplicative if and only if D is integrally closed. If the v-finite v-ideals of D are v-invertible, the v-content function c(·)_{v} can be extended from polynomials to rational functions over K by c(f)_{v} := c(f_{1})_{v}×c(f_{2})_{v}^{−1} for all f := \frac{f_{1}}{f_{2}} , f_{1}, f_{2} ∈ D[X].

If the v-finite v-ideals of D are v-cancellative with respect to the v-multiplication, which is shown to be equivalent to the v-finite v-ideals being v-invertible, D is called a v-domain. The quotient group Λ_{v}(D) of all integral v-finite v-ideals is referred to as the Lorenzen v-group of D; Λ_{v}^{+}(D) denotes the semigroup of all integral v-ideals in Λ_{v}(D). If D is a v-domain, the set

D^{v} := \{\frac{f}{g} : f, g ∈ D[X] with c(f)_{v} ⊆ c(g)_{v}\}

of rational functions over K becomes an integral domain, the Kronecker function ring of D with respect to the v-operation. Any D^{v}-ideal generated by finitely many rational functions \frac{f_{1}}{g} , …, \frac{f_{m}}{g}, f_{i}, g ∈ D[X], is a principal ideal of the form \frac{f}{g} D^{v}, where f(X) = f_{1} · X^{r_{1}} + … + f_{m} · X^{r_{m}} denotes a polynomial in which the exponents r_{i} are such that in the representation of f no monomials of equal degree occur. This shows that D^{v} is actually a Bezout domain. It is easy to see that D^{v} ∩ K = D. With Λ(D^{v}) (resp. Λ^{+}(D^{v})) denoting the Lorenzen (semi)group of D^{v}, i. e. the (semi)group of all (integral) finitely generated, hence principal D^{v}-ideals, we get the isomorphisms of gcd-(semi)groups Λ_{v}(D) ≅ Λ(D^{v}) resp. Λ_{v}^{+}(D) ≅ Λ^{+}(D^{v}).

If the v-finite v-ideals of D even form a group by v-multiplication, i. e. if the vinverse of any v-finite v-ideal is v-finite again, D is called Prüfer v-multiplication domain (PVMD). In this case, the Kronecker function ring D^{v} is equal to the quotient ring of D[X] with respect to the multiplicatively closed set N_{v}(D) of v-primitive polynomials

D_{v}(X) = D[X]_{Nv(D)} = \{\frac{f}{g} : f ∈ D[X], g ∈ N_{v}(D)\}.

We call D_{v}(X) the Nagata domain of D with respect to the v-operation.

If D is a Prüfer v-multiplication domain in which every v-ideal is v-finite, D is shown to be a Krull domain (and vice versa). Observing that the v-finiteness condition means that v-operation and t-operation coincide, we obtain ideal theoretic characterizations of a Krull domain which are analogue to those of a Dedekind domain. To be precise, D is a Krull domain if and only if any of the following conditions holds: (1) The fractional t-ideals form a group by t-multiplication. (2) Any t-ideal can be uniquely written as the t-product of finitely many prime t-ideals. (3) D is integrally closed, every ascending chain of t-ideals becomes stable (t-noetherian property) and any prime t-ideal is maximal in the set of all t-ideals.

We recall that a Krull domain is nothing else than a ring with divisor theory, i. e. a domain together with a factorial semigroup 𝔇^{+} and a homomorphism (·) : D^{∗} → 𝔇^{+} such that the following conditions hold: (D 1) a | b with respect to D ⇔ (a) | (b) with respect to 𝔇^{+}. (D 2) {d ∈ K : 𝔞 | (d)} = {d ∈ K : 𝔟 | (d)} ⇔ 𝔞 = 𝔟 for all 𝔞, 𝔟 ∈ 𝔇^{+}. According to this defintion the second ideal theoretic criterion for Krull domains cited above says that an adequate system of divisors is given by the factorial group of t-ideals, i. e. by the Lorenzen v-group Λ_{v}(D). Up to isomorphism, there is no other divisor theory for D than the homomorphism (·)_{v} : D^{∗} → Λ_{v}^{+}(D) with a ↦ (a)_{v} = aD.

An alternative to make the divisor theory for D explicit is given by the Kronecker function ring D^{v} of D with respect to the v-operation. It is well-known that D is a Krull domain if and only if its corresponding Nagata domain D_{v}(X) is a principal ideal domain. Since D_{v}(X) = D^{v} in this case, the order preserving isomorphism Λ_{v}(D) ≅ Λ(D^{v}) becomes an isomorphism of factorial groups. Thus, [ ·]_{v} : D^{∗} → Λ^{+}(D^{v}) with a ↦ [ a]_{v} := aD^{v} defines another divisor theory for D. It is called Kronecker’s divisor theory for D. The elements of D^{v} are referred to as integral Kronecker divisors, the elements in K(X) as fractional Kronecker divisors. Since any element in D^{v} is associated to a polynomial over K, we can think of Kronecker divisors as polynomials. Note that with respect to D^{v} every polynomial in K[X] can be interpreted as a greatest common divisor of its coefficients.

If T is the integral closure of a Krull domain D in the finite field extension L/K, then T is again a Krull domain whose Kroencker function ring T^{v} is the integral closure of D^{v} in the finite field extension L(X)/K(X). Thus, the equalities

T^{v} = T_{v}(X) = T [X]_{Nv(T)} = T [X]_{Nv(D)}

hold. Since the elements of D^{v} are contained in T^{v}, Kronecker divisors for D can be regarded as Kronecker divisors for T in a most natural way.

In regard to v-ideal divisors it is well-known that 𝔡_{v} = (𝔡T)_{v} ∩ K for all D-ideals 𝔡 (cf. [5, Satz 9]). This, however, implies that nothing is lost when we lift v-ideals of D to v-ideals of T. Thus, we may identify the v-ideals of D with the v-ideals they generate in T. Note that, in particular, the gcd-property of a Kronecker resp. v-ideal divisor remains untouched when passing on from one algebraic extension field to another. Informally speaking, “gcd in D stays gcd in T ”.

Since the algebraic structure of the field extension L(X)/K(X) mimicks that of L/K, we have [L(X) : K(X)] = [L : K] = m. In particular, the degrees of separability s and inseparability i are the same. Since every conjugation map over K is equal to the restriction of a conjugation map over K(X), we will denote the K-monomorphisms from L into the normal closure of L/K resp. the K(X)-monomorphism from L(X) into the normal closure of L(X)/K(X) uniformly by σ_{1}, …, σ_{s}. For the norm function of L/K resp. L(X)/K(X), which is given by ∏_{k=1}^{s} σ_{k}(·)^{i}, we will always use the abbreviated notation N_{L/K}, even when applied to L(X).

According to this definition the meaning of the norm of a Kronecker divisor is obvious, at least if it is referred to as an element of L(X). For an element of Λ(T^{v}), i. e. for a principal ideal of the form F T^{v}, F ∈ L(X), we define N_{L/K}(F T^{v}) := N_{L/K}(F)D^{v}. Since the map Λ(T^{v}) → Λ_{v}(T) with F T^{v} ↦ c(F)_{v} is a well-defined isomorphism and since any v-ideal of T can be written as the v-content of a polynomial with coefficients in L, this definition induces a well-defined norm function for v-ideal divisors of T by

N_{L/K}^{(v)}(𝔄) := c(N_{L/K}(F))_{v} for all 𝔄 ∈ Λ_{v}(T), F ∈ L(X) with 𝔄 = c(F)_{v}. It is called the v-ideal norm or just v-norm with respect to L/K. Note that per definitionem the v-norm of a v-ideal of T gives a v-ideal of D.

Proposition 1.1. Let L/K be an algebraic field extension with degree of separability s and degree of inseparability i. Let further 𝔄 = 𝔄_{v} be a v-ideal of T. If σ_{k}(𝔄)_{v}, 1 ⩽ k ⩽ s, is the v-ideal generated by σ_{k}(𝔄) in the normal closure of L/K, we get

N_{L/K}^{(v)}(𝔄) = σ_{1}(𝔄)_{v}^{i}× … ×σ_{s}(𝔄)_{v}^{i} ∩ K.

Thus, if L/K is a separable finite extension, the v-norm of a v-ideal is just the v-product of its conjugates.

Proof: It has already been mentioned that in case of an arbitary algebraic field extension L/K the equality 𝔡_{v} = (𝔡T)_{v} ∩ K holds for any D-ideal 𝔡. This, however, implies that nothing is lost when we lift v-ideals in L to v-ideals in the normal closure of L/K. Thus, we may assume without loss of generality that the finite extension L/K is normal. Let F ∈ L[X] be a polynomial with v-content 𝔄. Per definitionem, N_{L/K}^{(v)}(𝔄) = c(N_{L/K}(F))_{v}. In addition, we obtain

σ_{k}(𝔄)_{v} = σ_{k}(𝔄) = σ_{k}(c(F)_{v}) = c(σ_{k}(F))_{v} for 1 ⩽ k ⩽ s

since L/K is normal. Observing that the v-content in an integrally closed domain is always multiplicative we obtain

N_{L/K}^{(v)}(𝔄) = c(σ_{1}(F)^{i} · . . . · σ_{s}(F)^{i})_{v}
= c(σ_{1}(F))_{v}^{i}× … ×c(σ_{m}(F))_{v}^{i} ∩ K
= σ_{1}(c(F)_{v})^{i}× … ×σ_{m}(c(F)_{v})^{i} ∩ K
= σ_{1}(𝔄)^{i}× … ×σ_{m}(𝔄)^{i} ∩ K.  □

Proposition 1.2. Let F ∈ L(X) be a Kronecker divisor with v-content 𝔄 = 𝔄_{v}. If “∼” means “is associated to”, we have

N_{L/K}(F) ∼ gcd({N_{L/K}(α)}_{α∈L, F |α}).

In terms of v-ideals we obtain

N_{L/K}^{(v)}(𝔄) = ({N_{L/K}(α)}_{α∈𝔄})_{v}.

Proof: Making the divisor group of T explicit by the Lorenzen v-group we have gcd({α_{i}}_{i∈I}) = ({α_{i}}_{i∈I})_{v} for any family {α_{i}}_{i∈I} of elements in L (cf. [7, Satz 2.6]). Because of the definition of the v-norm it is therefore sufficient to prove the statement for Kronecker divisors. Obviously, N_{L/K}(F) | N_{L/K}(α) for all α ∈ L with F | α. Thus, it remains to show gcd({N_{L/K}(α)}_{α∈L,F |α}) | N_{L/K}(F). Let p ∈ D^{v} be a Kronecker divisor prime in D^{v}. Since T is a Krull domain, there exists a Kronecker divisor G_{p} ∈ T^{v} with gcd(p, G_{p}) ∼ 1 in T^{v} and F · G_{p} = α_{p} ∈ T (cf. [7, Satz 3.1, (G 2.4)]). The condition gcd(p, G_{p}) ∼ 1 in T^{v} implies that the corresponding norms are relatively prime in D^{v}, hence gcd(p, N_{L/K}(G_{p})) ∼ 1 in D^{v}. With v_{p}(·) denoting the p-exponent of a Kronecker divisor we obtain

v_{p}(N_{L/K}(α_{p})) = v_{p}(N_{L/K}(F) · N_{L/K}(G_{p}))
= v_{p}(N_{L/K}(F)) + v_{p}(N_{L/K}(G_{p}))
= v_{p}(N_{L/K}(F)).

Since min {v_{p}(N_{L/K}(α)): α ∈ L, F | α} ≤ v_{p}(N_{L/K}(α_{p})) = v_{p}(N_{L/K}(F)) for all prime Kronecker divisors p ∈ D^{v}, it follows that

gcd({N_{L/K}(α)}_{α∈L, F |α}) = ∏_{p∈D^{v} prime} p^{min{v_{p}(N_{L/K}(α)): α ∈ L, F | α}}

divides N_{L/K}(F). □

We want to end this section by proving that in the case of a Krull domain D the ismorphism Λ_{v}(D) ≅ Λ(D^{v}) is given by extending divisorial D-ideals to principal D^{v}-ideals and by contracting principal D^{v}-ideals to divisorial D-ideals. This follows immediately from

Lemma 1.3. Let D be a v-domain in which the v-operation is of finite type. Then 𝔞D^{v} = 𝔞_{v}D^{v} for all D-ideals 𝔞. In particular, fD^{v} = c(f)_{v}D^{v} for all f ∈ K[X].

Proof: Due to 𝔞 ⊆ 𝔞_{v} the inclusion 𝔞D^{v} ⊆ 𝔞_{v}D^{v} is trivial. Let α ∈ 𝔞_{v}D^{v}. There are α_{1}, …, α_{m} ∈ 𝔞_{v} and f_{1}, …, f_{m} ∈ D^{v} such that α = α_{1}f_{1} + … + α_{m}f_{m}. Since the v-operation is of finite type, for every α_{i} there is a finitely generated D-ideal 𝔞^{(i)} with 𝔞^{(i)} ⊆ 𝔞 and α_{i} ∈ 𝔞_{v}^{(i)}. Let 𝔟 be the D-ideal generated by 𝔞^{(1)}, …, 𝔞^{(m)}. Then, we have 𝔟 ⊆ 𝔞 and α_{i} ∈ 𝔟_{v} for 1 ⩽ i ⩽ m. Since 𝔟 is finitely generated, there is a polynomial f ∈ K[X] whose coefficients generate 𝔟. This implies α_{i} ∈ c(f)_{v}, hence f | α_{i} with respect to D^{v}. Thus, f | α, i. e. α ∈ fD^{v} = 𝔟D^{v} ⊆ 𝔞D^{v}. □

Theorem 1.4. Let D be a Krull domain with field of fractions K and T the integral closure of D in the finite field extension L/K. Then the following diagrams are commutative with the horizontal maps being inverse isomorphisms:

\begin{matrix}Λ_{v}(T) & ⟶^{· T^{v}} ; ⟵_{∩ L} & Λ(T^{v}) \\ ^{N_{L/K}^{(v)}} ↓ & & ↓^{N_{L/K}} \\ Λ_{v}(D) & ⟶^{· D^{v}} ; ⟵_{∩ K} & Λ(D^{v}) \end{matrix}  \begin{matrix}Λ_{v}(T) & ⟶^{· T^{v}} ; ⟵_{∩ L} & Λ(T^{v}) \\ ^{∩ K}↓ & & ↓^{∩ K(X)} \\ Λ_{v}(D) & ⟶^{· D^{v}} ; ⟵_{∩ K} & Λ(D^{v}) \end{matrix}

Proof: Since D and T are Krull domains, the corresponding Kronecker function rings D^{v} and T^{v} are principal ideal domains and all divisorial ideals of D resp. T are v-finite. The map Λ(D^{v}) → Λ_{v}(D), fD^{v} ↦ c(f)_{v}, is known to be a well-defined isomorphism with inverse map 𝔞_{v} ↦ 𝔟D^{v}, where 𝔟 is a finitely generated D-ideal with 𝔞_{v} = 𝔟_{v}. Since c(f)_{v} = fD^{v} ∩ K for all f ∈ K[X], the isomorphism fD^{v} ↦ c(f)_{v} is actually given by contracting D^{v}-ideals to divisorial D-ideals. Vice versa, the isomorphism 𝔞_{v} ↦ 𝔞D^{v} is given by extending divisorial D-ideals to D^{v}-ideals because 𝔞D^{v} = 𝔞_{v}D^{v} according to Lemma 1.3. By definition of the v-norm the first diagram is obviously commutative. Let F ∈ L[X] be a Kronecker divisor of T and 𝔄 the corresponding v-ideal of T such that 𝔄 = c(F)_{v} = F T^{v} ∩ L resp. 𝔄T^{v} = F T^{v}. Since the contracted ideal F T^{v} ∩ K(X) is a principal D^{v}-ideal, there is an f ∈ K[X] with fD^{v} = F T^{v} ∩ K(X). The corresponding v-ideal of D is given by fD^{v} ∩ K = c(f)_{v}, and we obtain F T^{v} ∩ K = F T^{v} ∩ L ∩ K = 𝔄 ∩ K = c(f)_{v}. This proves that the contraction of a divisorial T -ideal gives a divisorial D-ideal. This means that the map Λ_{v}(T) ⟶^{∩ K} Λ_{v}(D) is defined and that the second diagram is commutative. □

For the reader familiar with the theory of ∗-operations and ∗-ideals it is a wellknown fact that the following conditions are equivalent for a ∗-operation (·)_{∗} of finite type which is endlich arithmetisch brauchbar, i. e. for which the Kronecker function ring D^{(∗)} resp. the Lorenzen ∗-group is defined: (1) The ∗-finite ∗-ideals form a group by ∗-multiplication. (2) The map 𝔞_{∗} ↦ 𝔞_{∗}D^{(∗)} from the semigroup of ∗-ideals of D to the semigroup of D^{(∗)}-ideals is an isomorphism (cf. [4, II, § 3.1, Théorème 3]). In addition, it is easy to prove the ∗-theoretic analogue to Lemma 1.3, which says that 𝔞_{∗}D^{(∗)} ∩ K = 𝔞_{∗} for any ∗-ideal 𝔞_{∗} of D if the ∗-finite ∗-ideals are ∗-invertible. If any of the conditions (1) or (2) hold it can be deduced that the inversve map of 𝔞_{∗} ↦ 𝔞_{∗}D^{(∗)} is given by contracting D^{(∗)}-ideals to ∗-ideals of D. Translating this general result to Krull domains we get an alternative proof of Theorem 1.4.

Notation: To make the reading of this paper easier we will use the following standard of notation: Elements in K are denoted by small Latin letters (a, b, c, …), elements in the algebraic extension field L by small Greek letters (α, β, γ, …). D-ideals are denoted by small, T -ideals by capital German letters (𝔞, 𝔟, 𝔠, … resp. 𝔄, 𝔅, ℭ, …). Kronecker divisors of D, i. e. elements of the function field K(X), are denoted by small, Kronecker divisors of T by capital Latin letters (f, g, h, … resp. F, G, H, …). v-Ideals and Kronecker divisors that correspond to each other according to Theorem 1.4 are denoted by corresponding letters: 𝔞_{v} ↔ f, i. e. 𝔞_{v} = c(f)_{v}, 𝔟_{v} ↔ g, … resp. 𝔄_{v} ↔ F, 𝔅_{v} ↔ G, . . .. For D- resp. T -ideals extended to D_{v}(X) resp. T_{v}(X) we will also write 𝔞_{v}(X), 𝔟_{v}(X), … resp. 𝔄_{v}(X), 𝔅_{v}(X), …, i. e. we will set 𝔞_{v}(X) := 𝔞_{v}D^{v} = 𝔞_{v}D_{v}(X) etc. 𝔞_{v}[X], 𝔟_{v}[X], … resp. 𝔄_{v}[X], 𝔟_{v}[X] denote the set of polynomials in the indeterminate X with coefficients in 𝔞_{v}, 𝔟_{v}, … resp. 𝔄_{v}, 𝔅_{v}, …, i. e. we will set 𝔞_{v}[X] := 𝔞_{v}D[X] etc. Prime ideals or prime Kronecker divisors are denoted by the letter p in its adequate form (𝔭, 𝔓 resp. p, P). If not otherwise stated, all ideals in this paper are v-ideals so that the index letter v will usually be omitted.

2 Residue class fields, inertia degree and ramifaction index

Let D be a Krull domain with field of fractions K and 𝔞 = 𝔞_{v} ∈ Λ_{v}^{+}(D) an integral v-ideal of D that is not v-primitive. Extending the canonical epimorphism D → D/𝔞 to the ring of polynomials over D in the indeterminate X we obtain

\begin{aligned}φ_{𝔞} : & D[X] & ⟶ & D/𝔞 [X] \\ & a_{0} + a_{1}X + … + a_{m}X^{m} & ⟼ & \overline{a_{0}} + \overline{a_{1}}X + … + \overline{a_{m}}X^{m}, \end{aligned} (2.1)

where the bar stands for taking the residue modulo 𝔞, i. e. \overline{a_{i}} := a_{i} mod 𝔞. We are interested in the image of the set of v-primitive polynomials over D.

Proposition 2.2. The canonical epimorphism φ_{𝔞} : D[X] → D/𝔞 [X] of (2.1) maps the v-primitive polynomials of D to the non-zero divisors of D/𝔞 [X]. To be precise,

φ_{𝔞}(N_{v}(D)) = {φ_{𝔞}(f) : f ∈ D[X]^{∗}; fg ∈ 𝔞(X) for g ∈ D[X] ⇔ g ∈ 𝔞(X)}.

Proof: “⊆” Let g ∈ N_{v}(D) and h ∈ D[X] be polynomials over D with φ_{𝔞}(g) · φ_{𝔞}(h) = \overline{0}, i. e. gh ∈ 𝔞[X]. Since c(g)_{v} = D, it follows c(h)_{v} = c(g)_{v}×c(h)_{v} = c(gh)_{v} ⊆ 𝔞. This, however, means h ∈ 𝔞[X], hence φ_{𝔞}(h) = \overline{0}, which proves that φ_{𝔞}(g) is a non-zero divisor in D/𝔞 [X].

“⊇” Let φ_{𝔞}(h) be a non-zero divisor of D/𝔞 [X]. First, we prove that the videal generated by 𝔞 and the v-content of h is the unit ideal, i. e. (c(h)_{v}, 𝔞)_{v} = D. Assume that (c(h)_{v}, 𝔞)_{v} ⫋ D. By the v-invertiblity of any v-ideal in D, there is a fractional v-ideal 𝔟 = 𝔟_{v} with D ⫋ 𝔟_{v} such that 𝔟_{v}×(c(h)_{v}, 𝔞)_{v} = D. This implies (𝔞_{v}×𝔟_{v})×(c(h)_{v}, 𝔞)_{v} = 𝔞_{v} with 𝔞_{v}×𝔟_{v} ⫌ 𝔞_{v}. Since every v-ideal in a Krull domain is v-finite, there is a polynomial f ∈ K[X] with v-content 𝔞_{v}×𝔟_{v}. It follows

c(f)_{v}×(c(h)_{v}, 𝔞)_{v} = (c(f · h)_{v}, 𝔞 · c(f)_{v})_{v} = 𝔞_{v} = 𝔞,

hence fh ∈ 𝔞[X]. Since f, h ∉ 𝔞[X], we obtain φ_{𝔞}(f) · φ_{𝔞}(h) = \overline{0} with φ_{𝔞}(f), φ_{𝔞}(h) ≠ \overline{0}. This shows that φ_{𝔞}(f) is a zero divisor in D/𝔞 [X], which contradicts our assumption. Thus, (c(h)_{v}, 𝔞)_{v} = D. Recalling that any v-ideal in a Krull domain can be v-generated by two elements we conclude that there must be elements a, b ∈ 𝔞 such that (c(h)_{v}, a, b)_{v} = D. Let g = h + aX^{m} + bX^{m+1} with m := 1 + deg h. Then g is v-primitive, and from g − h = aX^{m} + bX^{m+1} ∈ 𝔞[X] we get φ_{𝔞}(g) − φ_{𝔞}(h) = \overline{0}. Thus, φ_{𝔞}(g) = φ_{𝔞}(h), which proves that there is a v-primitive polynomial that is mapped onto the non-zero divisor φ_{𝔞}(h). □

Proposition 2.3. Let D be a Krull domain and 𝔞 = 𝔞_{v} a v-ideal which is not vprimitive. Writing D/𝔞 [X]_{φ𝔞(Nv(D))} for the quotient ring of D/𝔞 [X] with respect to the multiplicative set of non-zero divisors φ_{𝔞} (N_{v}(D)), we have the isomorphisms

D_{v}(X)/𝔞(X) ≅ D[X]_{Nv(D)}/𝔞D[X]_{Nv(D)} ≅ D/𝔞 [X]_{φ𝔞(Nv(D))}.

Proof: We prove that the well-defined epimorphism

\begin{aligned}φ̃_{𝔞} : & D[X]_{Nv(D)} & ⟶ & D/𝔞 [X]_{φ𝔞(Nv(D))} \\ & \frac{f}{g} & ⟼ & φ_{𝔞}(f) · φ_{𝔞}(g)^{−1}, & f ∈ D[X], g ∈ N_{v}(D) \end{aligned}

has kernel 𝔞(X) = 𝔞D[X]_{Nv(D)}. Let h = \frac{f}{g}, f ∈ D[X], g ∈ N_{v}(D), be a Kronecker divisor of D. Since φ_{𝔞}(g) ≠ \overline{0}, we have

φ̃_{𝔞}(h) = \overline{0} ⇔ φ_{𝔞}(f) = \overline{0} ⇔ f ∈ 𝔞[X] ⇔ h ∈ 𝔞D[X]_{Nv(D)}.  □

Let 𝔭 be a prime v-ideal of D and p the corresponding Kronecker divisor, i. e. an element of K(X) with c(p)_{v} = 𝔭. Note that 𝔭 need not be a maximal D-ideal and consequently the residue class ring of D modulo 𝔭, which is an integral domain, need not be a field. We call the quotient field of D/𝔭 the residue class field of D with respect to 𝔭. It is denoted by K_{𝔭}. This terminology is obviously compatible with the one developed for Dedekind domains. Passing on to the Kronecker function ring D^{v} = D_{v}(X) = D[X]_{Nv(D)} of D the situation becomes even simpler and clearer. According to Theorem 1.4 the extended D_{v}(X)-ideal 𝔭D_{v}(X) = 𝔭(X) = pD_{v}(X) gives a prime Kronecker divisor of D. Since principal prime ideals are maximal, the residue class ring D_{v}(X)/pD_{v}(X) actually is a field, namely the residue class field of D_{v}(X) with respect to p, denoted by K(X)_{p}.

Theorem 2.4. Let D be a Krull domain with quotient field K and T the integral closure of D in a finite field extension L/K. Let p be a prime Kronecker divisor of D with v-content 𝔭 and P a prime Kronecker divisor of T with v-content 𝔓, such that P | p in T_{v}(X), i. e. 𝔓 ∩ K = 𝔭. Then

(1) K_{𝔭}(X) ≅ K(X)_{p} and L_{𝔓}(X) ≅ L(X)_{P}.

(2) [L_{𝔓} : K_{𝔭}] = [L(X)_{P} : K(X)_{p}] ⩽ L/K.

Proof: Ad (1) Since D/𝔭 is an integral domain, the corresponding polynomial ring D/𝔭 [X] is an integral domain as well. Thus, D/𝔭 [X] contains no zero divisors, which implies φ_{𝔭} (N_{v}(D)) = D/𝔭 [X] according to Proposition 2.2. Therefore,

D/𝔭 [X]_{φ𝔭(Nv(D))} = Quot(D/𝔭 [X]) = Quot(D/𝔭)(X) = K_{𝔭}(X).

The isomorphism now follows from Proposition 2.3.

Ad (2) By (1) the equality [L_{𝔓} : K_{𝔭}] = [L(X)_{P} : K(X)_{p}] is obvious. It is wellknown that [L_{𝔓} : K_{𝔭}] ⩽ [L : K] holds if D and T are Dedekind domains. Since D_{v}(X) and T_{v}(X) are even principal ideal domains we get [L(X)_{P} : K(X)_{p}] ⩽ [L(X) : K(X)] = [L : K]. □

In line with the usual terminology the inertia degree of the prime Kronecker divisor P with respect to L/K, i. e. the degree [L_{𝔓} : K_{𝔭}] = [L(X)_{P} : K(X)_{p}], is denoted by f_{L/K}(P). For the ramifaction index of P with respect to L/K we will write e_{L/K}(P). The meaning of the terms inertia degree and ramifacation index for a prime v-ideal in T is obvious. From the theory of Dedekind domains we obtain the following results.

Proposition 2.5. Let p ∈ D_{v}(X) be a prime Kronecker divisor of D whose prime factorization in T_{v}(X) is given by p ∼ P_{1}^{e_{1}} · . . . · P_{n}^{e_{n}}, where the P_{i} ∈ T_{v}(X) are distinct prime divisors with e_{L/K}(P_{i}) = e_{i}. Interpreting the residue class ring T_{v}(X)/pT_{v}(X) as a vector space over the residue class field D_{v}(X)/pD_{v}(X) = K(X)_{p} of dimension d we obtain:

(1) d = ∑_{i=1}^{n} e_{L/K}(P_{i}) · f_{L/K}(P_{i}) ⩽ [L : K]. If L/K is a separable extension, equality holds.

(2) If L/K is a normal extension, then e_{L/K}(P_{i}) = e and f_{L/K}(P_{i}) = f for all 1 ⩽ i ⩽ n. Thus, d = e · f · n ⩽ [L : K].

(3) If L/K is a Galois extension, then d = e · f · n = [L : K]

Proposition 2.6. Let L/K be a finite separable extension. If P ∈ T_{v}(X) and p ∈ D_{v}(X) are prime Kronecker divisors such that P | p with respect to T_{v}(X) and if 𝔓 and 𝔭 are the corresponding v-ideals, we have

N_{L/K}(P) ∼ p^{f_{L/K}(P)} and N_{L/K}^{(v)}(𝔓) = (𝔭^{f_{L/K}(𝔓)})_{v}.

3 Differents, discriminants and fundamental divisors

From now on let L/K be a finite separable extension of degree m. Then L(X)/K(X) is also separable of degree m. We denote the trace function with respect to L/K resp. L(X)/K(X) uniformly by T r_{L/K}. For A ⊆ L

A_{T/D} := {γ ∈ L : T r_{L/K}(γA) ⊆ D}

is called the complementary set corresponding to A. Any complementary set is a Dmodule. If A is a free module with D-basis {ω_{1}, …, ω_{m}}, then A_{T/D} is a free D-module with the complementary basis {ω̂_{1}, …, ω̂_{m}}. The complementary basis is determined by T r_{L/K}(ω_{i}ω̂_{j}) = δ_{ij}, 1 ⩽ i, j ⩽ m, where δ_{ij} means the Kronecker symbol. If A = 𝔄 is a fractional T -ideal, the complementary set 𝔄_{T/D} is a fractional T -ideal, too. If T is a Krull domain, it even is a v-ideal. This follows from

Lemma 3.1. Let 𝔄 be an arbitrary fractional ideal of the Krull domain T. Then,

𝔄(X)_{Tv(X)/Dv(X)} ∩ L = 𝔄_{T/D}.

Proof: Since any element in T_{v}(X) can be written as the quotient of a polynomial F over T and a v-primitive polynomial g over D and since T r_{L/K}(\frac{F}{g}) = \frac{1}{g} · T r_{L/K}(F) for all F ∈ T [X] and g ∈ D[X], we have

𝔄(X)_{Tv(X)/Dv(X)} ∩ L

= {γ ∈ L(X) : T r_{L/K}(γ · 𝔄(X)) ⊆ D_{v}(X)} ∩ L

= \{γ ∈ L : T r_{L/K}(γ · \frac{F}{g} ) ∈ D_{v}(X) for all F ∈ 𝔄[X], g ∈ N_{v}(D)\}

= {γ ∈ L : T r_{L/K}(γ · F) ∈ D_{v}(X) ∩ K[X] = D[X] for all F ∈ 𝔄[X]}

= {γ ∈ L : T r_{L/K}(γ · α) ∈ D for all α ∈ 𝔄} = 𝔄_{T/D}.  □

Proposition 3.2. If T is a Krull domain, the complementary ideal of an arbitrary T -ideal is always a v-ideal. Two T -ideals generating the same v-ideal have the same complementary ideal.

Proof: Let 𝔄 be an arbitrary T -ideal. By Theorem 1.4 the contracted T -ideal 𝔄_{v}(X)_{Tv(X)/Dv(X)} ∩ L = 𝔄_{T/D} is a v-ideal. If 𝔅 is another T -ideal with 𝔄_{v} = 𝔅_{v}, we have 𝔄(X) = 𝔄_{v}(X) = 𝔅_{v}(X) = 𝔅(X) by Lemma 1.3. Now Lemma 3.1 yields

𝔄_{T/D} = 𝔅_{T/D}. □

The different of the integral ring extension T/D is given by the integral v-ideal (T_{T/D})^{−1} and is denoted by 𝔇_{T/D}. Correspondingly, 𝔇_{v}(X)_{T/D} := (T_{v}(X)_{Tv(X)/Dv(X)})^{−1} means the different of the integral ring extension T_{v}(X)/D_{v}(X). By Lemma 3.1 we have T_{T/D} = T_{v}(X)_{Tv(X)/Dv(X)} ∩ L. Theorem 1.4, however, says that contracting T_{v}(X)-ideals to T -ideals is an isomorphism. Thus, we obtain

Proposition 3.3. Let T be a Krull domain. Then 𝔇_{T/D} = 𝔇_{v}(X)_{T/D} ∩ L.

If T is a Krull domain, all we know about differents in Dedekind domains can be applied to the Kronecker function ring T^{v} = T_{v}(X), which is a principal ideal domain in this case. Translating the results into the language of divisorial ideals we see that, according to Theorem 1.4 and Proposition 3.3, most of the well-known ideal theoretic statements on differents in Dedekind domains, such as Dedekind’s Different theorem or the Different tower theorem, carry over to differents in Krull domains.

The same is true for discriminants in Krull domains. It is well known that the discriminant of an extension T/D is the ideal norm of the corresponding different if D resp. T are Dedekind domains. The discriminant of T_{v}(X)/D_{v}(X) is therefore given by the integral Kronecker divisor 𝔡_{v}(X)_{T/D} := N_{L/K}(𝔇_{v}(X)_{T/D}). Correspondingly, we define the discriminant of T/D to be the integral v-ideal 𝔡_{T/D} := N_{L/K}^{(v)}(𝔇_{T/D}).

Theorem 1.4 and Proposition 3.3 yield

𝔡_{T/D} = 𝔡_{v}(X)_{T/D} ∩ L.

Thus, the well-known ideal theoretic statements on discriminants in Dedekind domains, such as Dedekind’s discrimiant theorem or the Discriminant tower theorem, remain valid for Krull domains.

In fact, it would be sufficient to know all the results on differents and discriminants in the case of an integral extension of principal ideal domains instead of Dedekind domains. We leave it to the reader to simplify the standard proofs. It is most likely that many proofs do without the methods of localization and completion, which are usually applied.

There is another point which makes things easier in Kronecker’s divisor theory. If the finite extensions L/K resp. L(X)/K(X) are separable, the principal ideal domain T_{v}(X) becomes a free D_{v}(X)-module of degree [L(X) : K(X)] = m (cf. [8, V, § 4, Corollary 2 of Theorem 7]). In particular, the algebraic function field L(X) has an integral basis over K(X). This remarkable fact has not been put much emphasis on yet.

Let {F_{1}, …, F_{m}}, F_{i} ∈ T_{v}(X), be an integral basis of L(X) over K(X). Since every element in T_{v}(X) can be written as the quotient of a polynomial over T and a polynomial over D which is v-primitive, we may assume without loss of generality that the F_{i} are polynomials with coefficients in T. Let α_{1}, …, α_{r} be the coefficients of F_{1}, …, F_{m}. Any Kronecker divisor of T whose denominator is a polynomial with exactly these coefficients and whose nominator is v-primitive is called fundamental Kronecker divisor with respect to the integral basis {F_{1}, …, F_{m}}. The polynomials α_{1}X + … + α_{r}X^{r} and F := F_{1}X^{r_{1}} + … + F_{m}^{r_{m}} with r_{1} = 0 and r_{i+1} = i · (1 + max_{1⩽j⩽m}{deg f_{j}}) for 1 ⩽ i ⩽ m−1 are examples of fundamental Kronecker divisors. It will be sufficient to think of fundamental divisors as polynomials of exactly this type.

Note that any fundamental divisor is a greatest common divisor of its corresponding integral basis, but that not any greatest common divisor of an integral basis is fundamental. This is obvious since

T_{v}(X) = F_{1}D_{v}(X) + … + F_{m}D_{v}(X) ⊆ F_{1}T_{v}(X) + … + F_{m}T_{v}(X) ⊆ T_{v}(X), which means that the elements of an integral basis {F_{1}, …, F_{m}} are always relatively prime with respect to T_{v}(X).

To show that a fundamental divisor is also a primitive element of the separable extension L(X)/K(X) we have to take a look at the element different of F. Remember that if χ(t) denotes the characteristic polynomial of F over K(X), the element different of F is defined by

diff_{L/K}(F) = \frac{d}{dt}χ(t)|_{t=F} = \frac{d}{dt}N_{L/K}(t − F)|_{t=F} = ∏_{k=2}^{m} (F − σ_{k}(F)). (3.4)

Proposition 3.5. Let L/K be a finite separable extension and F ∈ L[X] a polynomial with the coefficients α_{1}, …, α_{r}. Then the following conditions are equivalent:

(1) diff_{L/K}(F) ≠ 0.

(2) L(X) = K(X)[F].

(3) L = K[α_{1}, …, α_{r}].

Proof: The equivalence (1) ⇔ (2) is well-known.

(1) ⇒ (3) Let diff_{L/K}(F) ≠ 0. Suppose K′ := K(α_{1}, …, α_{r}) ⫋ L, i. e. [L : K′] > 1, in particular. Since the field polynomial of F over K is given by χ(t) = N_{L/K}(t − F) = N_{L/K′}(N_{K′/K}(t − F)) = N_{K′/K}(t − F)^{[L:K′]},

diff_{L/K}(F) = [L : K′] · N_{L/K′}(F − F)^{[L:K′]−1} · \frac{d}{dt}N_{K′/K}(t − f)|_{t=F} = 0,

which is a contradiction.

(3) ⇒ (1) Let L = K(α_{1}, …, α_{r}). Assume diff_{L/K}(F) = 0. By (3.4) this means there is a k ∈ {2, …, m} such that σ_{k}(α_{i}) = α_{i} for all 1 ⩽ i ⩽ r. This, however, implies σ_{k}(α) = α for all α ∈ L, hence σ_{k} = σ_{1} = id_{L}, which is impossible. □

To show that the fundamental Kronecker divisor F with the coefficients α_{1}, …, α_{r} is a primitive element of the separable extension L(X)/K(X), hence a divisor with diff_{L/K}(F) ≠ 0, it is therefore sufficient to prove L = K[α_{1}, …, α_{r}]. Since

T_{v}(X) = F_{1}D_{v}(X) + … + F_{m}D_{v}(X) ⊆ D_{v}(X)[α_{1}, …, α_{r}] ⊆ T_{v}(X)

the Kronecker function ring T_{v}(X) is a finitely generated extension ring of D_{v}(X) with T_{v}(X) = D_{v}(X)[α_{1}, …, α_{r}]. Passing on to the corresponding quotient fields we obtain

L = L(X) ∩ L = K(X)(α_{1}, …, α_{r}) ∩ L = K(α_{1}, …, α_{r}).

Since the α_{i} are algebraic over K, this yields L = K[α_{1}, …, α_{r}].

We will show that even more is true, namely that any fundamental divisor F generates the Kronecker function ring T_{v}(X) over D_{v}(X), i. e. T_{v}(X) = D_{v}(X)[F]. This follows from

Theorem 3.6. Let D be a Krull domain with quotient field K, T the integral closure of D in the finite separable extension L/K and F ∈ T [X] a fundamental Kronecker divisor of T with respect to the integral basis {F_{1}, …, F_{m}}. If the residue class field extensions L(X)_{P}/K(X)_{p} are separable for all prime Kronecker divisors P ∈ T_{v}(X), p ∈ D_{v}(X) with P | p, we have

𝔇_{v}(X)_{T/D} = diff_{L/K}(F) T_{v}(X).

We prove this theorem by means of

Lemma 3.7. Let D be a Krull domain with quotient field K, T the integral closure of D in the finite separable extension L/K and F ∈ T [X] a fundamental Kronecker divisor of T with respect to the integral basis {F_{1}, …, F_{m}}. Then we have

diff_{L/K}(F) ∼ gcd({diff_{L/K}(G) : G ∈ T_{v}(X) with L(X) = K(X)[G]}).

Proof: Let α_{1}, …, α_{r} be the coefficients of F_{1}, …, F_{m}. Since we have the equality T_{v}(X) = D_{v}(X)[α_{1}, …, α_{r}] any element G ∈ T_{v}(X) can be written as

G = ∑_{i_{1},... ,i_{r}∈ℕ}d_{i_{1},... ,i_{r}}α_{1}^{i_{1}} · . . . · α_{r}^{i_{r}} with d_{i_{1},... ,i_{r}} ∈ D_{v}(X),

where only finitely many d_{i_{1},... ,i_{r}} are ≠ 0. Let G ∈ T_{v}(X) be a primitive element of the separable field extension L(X)/K(X), i. e. let diff_{L/K}(G) ≠ 0. We denote the normal closure of L(X)/K(X) by \overline{L(X)} and the integral closure of T_{v}(X) in \overline{L(X)} by \overline{T_{v}(X)}. Let σ be a K(X)-automorphism of \overline{L(X)}. To show diff_{L/K}(F) | diff_{L/K}(G) we first prove by induction on the number of coefficients r that G − σ(G) is a linear combination of α_{1} − σ(α_{1}), …, α_{r} − σ(α_{r}) over \overline{T_{v}(X)}. For r = 1 we have G = ∑_{i∈ℕ} d_{i} · α_{1}^{i} with d_{i} ∈ D_{v}(X). This, however, implies

G − σ(G) = ∑_{i∈ℕ}d_{i}(α_{1}^{i} − σ(α_{1})^{i})

= (α_{1} − σ(α_{1})) · ∑_{i∈ℕ}d_{i} · (α_{1}^{i−1} + α_{1}^{i−2}σ(α_{1}) + … + α_{1}σ(α_{1})^{i−2} + σ(α_{1})^{i−1}),

which obviously is an element of the desired form. Suppose now that the statement has been proved for r ⩾ 1 and let

G = ∑_{i_{1},... ,i_{r},i_{r+1}∈ℕ}d_{i_{1},... ,i_{r},i_{r+1}} · α_{1}^{i_{1}} · . . . · α_{r}^{i_{r}} · α_{r+1}^{i_{r+1}}

= ∑_{i_{r+1}∈ℕ}α_{r+1}^{i_{r+1}} · ∑_{i_{1},... ,i_{r}∈ℕ}d_{i_{1},... ,i_{r},i_{r+1}} · α_{1}^{i_{1}} · . . . · α_{r}^{i_{r}}.

We define G_{i_{r+1}} to be the Kronecker divisor ∑ d_{i_{1},... ,i_{r},i_{r+1}} · α_{1}^{i_{1}} · . . . · α_{r}^{i_{r}}. Then, by induction hypothesis, G_{i_{r+1}} − σ(G_{i_{r+1}}) is a linear combination of α_{1} − σ(α_{1}), …, α_{r} − σ(α_{r}) over \overline{T_{v}(X)}. Thus,

G − σ(G) = ∑_{i_{r+1}∈ℕ}(G_{i_{r+1}} · α_{r+1}^{i_{r+1}} − σ(G_{i_{r+1}}) · σ(α_{r+1}^{i_{r+1}}))

= ∑_{i_{r+1}∈ℕ}(α_{r+1}^{i_{r+1}}(G_{i_{r+1}} − σ(G_{i_{r+1}})) + σ(G_{i_{r+1}})(α_{r+1}^{i_{r+1}} − σ(α_{r+1}^{i_{r+1}})))

is a linear combination of α_{1} − σ(α_{1}), …, α_{r+1} − σ(α_{r+1}) over \overline{T_{v}(X)}, where we made use of the fact that

α_{r+1}^{i_{r+1}} − σ(α_{r+1}^{i_{r+1}}) = (α_{r+1} − σ(α_{r+1})) · (α_{r+1}^{i_{r+1}−1} + … + σ(α_{r+1}^{i_{r+1}−1}))

and α_{r+1}^{i+1} resp. σ(G_{i_{r+1}})(α_{r+1}^{i_{r+1}−1} + … + σ(α_{r+1}^{i_{r+1}−1})) are elements in \overline{T_{v}(X)}. For the fundamental divisor F = α_{1}X^{i_{1}} + … + α_{r}X^{i_{r}}, i_{k} ≠ i_{l} for k ≠ l, with respect to the integral basis {F_{1}, …, F_{m}} we have

F − σ(F) = (α_{1} − σ(α_{1}))X^{i_{1}} + … + (α_{r} − σ(α_{r}))X^{i_{r}}.

Since \overline{T_{v}(X)} is equal to the Kronecker function ring of \overline{T}, the Kronecker divisor F −σ(F) divides each of its coefficients in \overline{T_{v}(X)}. This however, implies F − σ(F) | G − σ(G) in \overline{T_{v}(X)} for any G ∈ T_{v}(X) with diff_{L/K}(G) ≠ 0. Since diff_{L/K}(F) = ∏_{σ≠id_{L}} (F − σ(F)), we obtain diff_{L/K}(F) | diff_{L/K}(G) in \overline{T_{v}(X)} ∩ L(X) = T_{v}(X). Thus, diff_{L/K}(F) is a greatest common divisor off all element differents diff_{L/K}(G) ≠ 0. □

Proof of Theorem 3.6: It is well-known that the different of a Dedekind domain is the greatest common divisor of all element differents ≠ 0 (cf. [6, III, § 2, Proposition 8] or [3, XII, Satz 4.3]). Thus, by Lemma 3.7

𝔇_{v}(X)_{T/D} = ({diff_{L/K}(G) : G ∈ T_{v}(X) with L(X) = K(X)[G]})T_{v}(X)
= diff_{L/K}(F) T_{v}(X).  □

Theorem 3.6 is indeed remarkable. Since we have

N_{L/K}(diff_{L/K}(G)) = (−1)^{\frac{1}{2} n(n−1)}discr_{L/K}(G)

for any primitive element G ∈ T_{v}(X) of the separable extension L(X)/K(X), it follows that the discriminant of T_{v}(X)/D_{v}(X) is given by

𝔡_{v}(X)_{T/D} = discr_{L/K}(F) D_{v}(X).

But we already know 𝔡_{v}(X)_{T/D} = discr_{L/K}(F_{1}, …, F_{m}) since the discriminant of the integral extension T_{v}(X)/D_{v}(X) is just the discriminant of any integral basis. Thus, {1, F, …, F^{m−1}} is shown to be an integral basis of L(X)/K(X), too, and we obtain

Theorem 3.8. Let D be a Krull domain with quotient field K and T the integral closure of D in the finite separable extension L/K. All residue class field extensions L(X)_{P}/K(X)_{p} are supposed to be separable for any pair of prime Kronecker divisors P ∈ T_{v}(X), p ∈ D_{v}(X) with P | p. If F ∈ T [X] is a fundamental Kronecker divisor of T, we have

T_{v}(X) = D_{v}(X)[F].

In particular, there are no common inessential discriminant divisors ≁ 1 with respect to the integral extension T_{v}(X)/D_{v}(X).

4 Factorizing primes in Kronecker’s divisor theory

Suppose now that the assumptions of Theorem 3.8 are fulfilled. Since we have T_{v}(X) = D_{v}(X)[F] for any fundamental Kronecker divisor F, we can always apply Kummer’s decomposition theorem (cf. [6, I, § 8, Proposition 25]) in order to factorize prime Kronecker divisors of D_{v}(X) in the integral closure T_{v}(X).

Let p ∈ D_{v}(X) be a prime Kronecker divisor of D_{v}(X). Assume that F ∈ D_{v}(X) is a fundamental divisor with minimal polynomial μ(t) ∈ D_{v}(X)[t]. Reduction modulo pD_{v}(X) gives the decomposition

\overline{μ(t)} = \overline{μ_{1}(t)}^{e_{1}} · . . . · \overline{μ_{n}(t)}^{e_{n}}

with μ_{i} ∈ D_{v}(X)[t] such that \overline{μ_{i}} is prime in D_{v}(X)/pD_{v}(X)[t]. Then the decomposition of p into irreducible factors in T_{v}(X) is given by

p ∼ P_{1}^{e_{1}} · . . . · P_{n}^{e_{n}}, where P_{i} ∼ gcd(p, μ_{i}(F)) with f_{L/K}(P_{i}) = deg μ_{i}(t) for all 1 ⩽ i ⩽ n.

Let us have a look at the classical situation of algebraic number theory, i. e. let D be the ring of rational integers ℤ with quotient field K = ℚ and T the ring of algebraic integers in the algebraic number field L. Then there is an integral basis {ω_{1}, …, ω_{m}} of L/K which is also an integral basis of L(X)/K(X). Thus, we can choose a fundamental Kronecker divisor of the form F = ω_{1}X + … + ω_{m}X^{m}. Note that in the case of an algebraic number field the v-operation is the identity on the system of all ideals. Thus, v-primitivity means primitivity in the usual sense, i. e. a polynomial over ℤ resp. T is v-primitive if and only if its coefficients generate the unit ideal. In ℤ this is obviously equivalent to the condition that all coefficients are relatively prime. Thus, the Kronecker function rings ℤ_{v}(X) and T_{v}(X) are given by ℤ_{v}(X) = \{\frac{f}{g} : f, g ∈ ℤ[X], gcd(coefficients of g) ∼ 1\} and T_{v}(X) = \{\frac{F}{G} : F, G ∈ T [X], (coefficients of G)T = T }.

Suppose that p ∈ ℤ is a rational prime with corresponding prime ideal 𝔭 = pℤ. Let P ∈ T_{v}(X) be a prime Kronecker divisor of T with corresponding v-ideal 𝔓 such that P | p in T_{v}(X). Since 𝔭 and 𝔓 are maximal ideals the residue class rings ℤ/𝔭 = ℤ_{p} and T/𝔓 are fields, namely the residue class fields of 𝔭 and 𝔓. Since ℤ_{p} is finite, T/𝔓 is separable over ℤ_{p}. This implies that T/𝔓 (X) is separable over ℤ_{p}(X). By Theorem 2.4.(1) it follows that T_{v}(X)/P T_{v}(X) is separable over ℤ_{v}(X)/pℤ_{v}(X). Thus, the conditions of Theorem 3.8 are fulfilled and we obtain T_{v}(X) = ℤ_{v}(X)[F].

Let μ(t) ∈ ℤ_{v}(X)[t] be the minimal polynomial of the fundamental divisor F. The proof of Proposition 2.3 shows that reducing a Kronecker divisor \frac{f}{g} ∈ ℤ_{v}(X), f ∈ ℤ[X], g ∈ N_{v}(ℤ), modulo pℤ_{v}(X) = 𝔭(X) actually means reducing the coefficients of f and g modulo pℤ = 𝔭. Since μ(t) has coefficients in ℤ[X] ⊆ ℤ_{v}(X), reduction modulo pℤ_{v}(X) gives the polynomial \overline{μ(t)} with coefficients in ℤ_{p}[X] ⊆ Quot(ℤ_{p}[X]) ≅ ℤ_{v}(X)/pℤ_{v}(X). Note that by Gauss’ Lemma any irreducible factor of \overline{μ(t)} in ℤ_{p}[X][t] remains irreducible over ℤ_{v}(X)/pℤ_{v}(X). Thus, it is sufficient to determine the prime decomposition of \overline{μ(t)} over ℤ_{p}[X]. Since ℤ_{p} is a finite field, this can be done in a finite number of steps. If μ_{1}, …, μ_{n} ∈ ℤ[X][t] are such that

\overline{μ(t)} = \overline{μ_{1}(t)}^{e_{1}} · . . . · \overline{μ_{n}(t)}^{e_{n}} with \overline{μ_{i}(t)} prime in ℤ_{p}[X][t],

all prime Kronecker divisors of T dividing p in T_{v}(X) are given by

P_{i} ∼ gcd(p, μ_{i}(F)) ∼ p + μ_{i}(F) · X,

and we obtain the prime decomposition of p in T_{v}(X) by

p ∼ P_{1}^{e_{1}} · . . . · P_{n}^{e_{n}}.

A similar result has also been obtained by H.M. Edwards in the case of algebraic number fields (cf. [1, §§ 2.4–2.7]). Edward’s presentation, however, is unfamiliar to the modern reader and sometimes lacks the clarity of modern mathematical language. In my opinion, it does not reveal the real cause why Kronecker divisors make Kummer’s factorization method work also for common inessential discriminant divisors.

5 Bibliography

[1] Harold M. Edwards: Divisor Theory. — Boston u. a.: Birkhäuser 1990.

[2] Harley Flanders: The meaning of the form calculus in classical ideal theory. — Transactions of the American Mathematical Society 95 (1960), 92–100.

[3] Karl-Bernhard Gundlach: Einführung in die Zahlentheorie. — Mannheim/Wien/ Zürich: Bibliographisches Institut 1972.

[4] Paul Jaffard: Systèmes Idéaux. — Paris: Dunod 1960.

[5] Wolfgang Krull: Beiträge zur Arithmetik kommutativer Integritätsbereiche. II. v-Ideale und vollständig ganz abgeschlossene Integritätsbereiche. — Mathematische Zeitschrift 41 (1936), 665–679.

[6] Serge Lang: Algebraic Number Theory. — New York u. a.: Springer 1996.

[7] Friedemann Lucius: Ringe mit einer Theorie des größten gemeinsamen Teilers. — Mathematica Gottingensis 7 (1997), 1–70.

[8] Oscar Zariski & Pierre Samuel: Commutative Algebra, Vol. 1. — New York/Heidelberg/Berlin: Springer 1958.