Dedekind's Independence Theorem

The independence set of Dedekind is a mathematical theorem that within the algebra is settled and the mathematician Richard Dedekind back. The theorem deals with the question of the linear independence of homomorphisms from semigroups into the unit groups of commutative bodies and as such leads to elementary structural theorems of Galois theory .

Formulation of the sentence

Following Kurt Meyberg's presentation , the sentence can be stated as follows:

Given are a ( multiplicative ) semigroup and a commutative field and, in addition, homomorphisms from into the Abelian group of units from . ${\displaystyle H\neq \emptyset }$ ${\displaystyle K}$ ${\displaystyle {\sigma }_{1},\ldots ,{\sigma }_{n}\;\;(n\in \mathbb {N} )}$ ${\displaystyle H}$ ${\displaystyle K^{*}}$ ${\displaystyle K}$

Then are equivalent:

(A1) They are different in pairs . ${\displaystyle {\sigma }_{1},\ldots ,{\sigma }_{n}}$

(A2) They form a linearly independent family of the function space . ${\displaystyle {\sigma }_{1},\ldots ,{\sigma }_{n}}$ ${\displaystyle K}$ ${\displaystyle \mathrm {Abb} (H,K)}$

Proof of the theorem

Based on Emil Artin and Kurt Meyberg, the following proof can be given:

A1 → A2

Here complete induction is carried out.

Induction start

Be it and with it . ${\displaystyle n=1}$ ${\displaystyle k_{1}\in K}$ ${\displaystyle k_{1}\sigma _{1}=0\in \mathrm {Abb} (H,K)}$

Then

${\displaystyle k_{1}\sigma _{1}(x)=0\;\;(\forall x\in H)}$ .

Because there are so one with ${\displaystyle H\neq \emptyset }$ ${\displaystyle x_{0}\in H}$

${\displaystyle k_{1}\sigma _{1}(x_{0})=0}$ .

Because of and the absence of zero divisors , then results ${\displaystyle \sigma _{1}(x_{0})\in K\setminus \{0\}}$ ${\displaystyle K}$

${\displaystyle k_{1}=0}$ .

Induction step

Let and be the statement already proven for homomorphisms of the kind described. ${\displaystyle n>1}$ ${\displaystyle n-1}$

Let any body elements be given and let it count in the equation ${\displaystyle n}$ ${\displaystyle k_{1},\ldots ,k_{n}}$ ${\displaystyle \mathrm {Abb} (H,K)}$

(a)   . ${\displaystyle \sum _{j=1}^{n}{k_{j}\sigma _{j}}=0}$

What is to be shown is that

(b) ${\displaystyle k_{1}=\ldots =k_{n}=0}$

applies.

First of all there is because of a with   . ${\displaystyle \sigma _{1}\neq \sigma _{n}}$ ${\displaystyle h_{0}\in H}$ ${\displaystyle \sigma _{1}(h_{0})\neq \sigma _{n}(h_{0})}$

This is now fixed. ${\displaystyle h_{0}\in H}$

Furthermore, (a) means that always

(c) ${\displaystyle 0=\sum _{j=1}^{n}{k_{j}\sigma _{j}(x)}\in K\;\;(\forall x\in H)}$

consists.

Since, because of the semigroup property, is also always for anything , (c) on the one hand leads to ${\displaystyle x\in H}$ ${\displaystyle hx\in H}$

(d) ${\displaystyle 0=\sum _{j=1}^{n}{k_{j}\sigma _{j}(h_{0}x)}=k_{1}\sigma _{1}(h_{0})\sigma _{1}(x)+\sum _{j=2}^{n}{k_{j}\sigma _{j}(h_{0})\sigma _{j}(x)}}$

and on the other hand too

(e)   . ${\displaystyle 0=\sigma _{1}(h_{0})\sum _{j=1}^{n}{k_{j}\sigma _{j}(x)}=k_{1}\sigma _{1}(h_{0})\sigma _{1}(x)+\sum _{j=2}^{n}{k_{j}\sigma _{1}(h_{0})\sigma _{j}(x)}}$

The subtraction of the equation (s) of the equation (d) results in

(f)   . ${\displaystyle 0=\sum _{j=2}^{n}{k_{j}{\bigl (}\sigma _{j}(h_{0})-\sigma _{1}(h_{0}){\bigr )}\sigma _{j}(x)}}$

Equation (f) holds for each and so one has in ${\displaystyle x\in H}$ ${\displaystyle \mathrm {Abb} (H,K)}$

(g)   . ${\displaystyle 0=\sum _{j=2}^{n}{k_{j}{\bigl (}\sigma _{j}(h_{0})-\sigma _{1}(h_{0}){\bigr )}\sigma _{j}}}$

Since according to the induction hypothesis the in are more than linearly independent, it follows from (g) ${\displaystyle \sigma _{2},\ldots ,\sigma _{n}}$ ${\displaystyle \mathrm {Abb} (H,K)}$ ${\displaystyle K}$

(H) ${\displaystyle k_{j}{\bigl (}\sigma _{j}(h_{0})-\sigma _{1}(h_{0}){\bigr )}=0\;\;(j=2,\ldots ,n)}$

and particularly

(i)   . ${\displaystyle k_{n}{\bigl (}\sigma _{n}(h_{0})-\sigma _{1}(h_{0}){\bigr )}=0}$

Because with (i) one also has ${\displaystyle \sigma _{n}(h_{0})-\sigma _{1}(h_{0})\neq 0}$

(j)   . ${\displaystyle k_{n}=0}$

Substituting (j) into (a) one then has the equation ${\displaystyle \mathrm {Abb} (H,K)}$

(k)   , ${\displaystyle \sum _{j=1}^{n-1}{k_{j}\sigma _{j}}=0}$

with which, if the induction hypothesis is applied again to the in over linearly independent then directly the equation ${\displaystyle \mathrm {Abb} (H,K)}$ ${\displaystyle K}$ ${\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}$

(l) ${\displaystyle k_{1}=\ldots =k_{n-1}=0}$

follows.

By connecting (j) and (l) , (b) is finally shown.

A2 → A1

Nothing further needs to be shown about this implication, since the vectors of a linearly independent family of every vector space are always different in pairs.

Inferences

1. Each family of pairwise different monomorphisms from one body to another is more than linearly independent. ${\displaystyle ({{\sigma }_{i}\colon \,K_{1}\to K_{2}})_{i\in I}}$ ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$ ${\displaystyle \mathrm {Abb} (K_{1},K_{2})}$ ${\displaystyle K_{2}}$
2. For every finite field extension the order of the Galois group is limited by the degree of the field extension : ${\displaystyle L/K}$

${\displaystyle |\mathrm {Gal} (L/K)|\leq [L\colon K]}$   .

Notes on naming

The independence set of Dedekind (or him closely related versions) encountered in the literature to algebra under different names. This is what BL van der Waerden calls it the independence principle . In Karpfinger -Meyberg, for example, Corollary 1 above (in the formulation for finite families) is called a Dedekind's lemma . A similar term can be found in English-language literature, for example in PM Cohn , who lists a closely related sentence as Dedekind's lemma ( German  dedekind's lemma ). From RBJT Allenby again he is called Dedekind's independence theorem ( German  Dedekind independence set called).

Related results

A related result, which also goes back to Dedekind, is the following:

Let and two commutative fields and further be a finite subgroup of the - automorphism group with as a fixed field . ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle \Gamma }$ ${\displaystyle L}$ ${\displaystyle \operatorname {Aut} (L)}$ ${\displaystyle K=\operatorname {Fix} (\Gamma )}$

Then is   . ${\displaystyle [L\colon K]=|{\Gamma }|}$

Karpfinger and Meyberg call the result Dedekind's theorem . In the English-language algebra literature, for example by PM Cohn, it is also known (with reference to the mathematician Emil Artin ) as Artin's theorem ( German  Artinian sentence ), whereby Cohn clarifies that the actual author is not Artin, but Dedekind.

Kurt Meyberg leads in his algebra. Part 2 also relies on this Artinian sentence , but it also gives another sentence by Emil Artin that is closely related to the result mentioned above, namely the following:

Let us be and two commutative fields and a finite field extension. ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle L/K}$

Then are equivalent:

(A) is a Galois expansion . ${\displaystyle L/K}$

(B)   . ${\displaystyle [L\colon K]=|\mathrm {Gal} (L/K)|}$

(C) is a normal and separable body extension at the same time . ${\displaystyle L/K}$

(D) is the decay field of a polynomial that is separable over . ${\displaystyle L/K}$ ${\displaystyle K}$

swell

• RBJT Allenby: Rings, Fields and Groups . An Introduction to Abstract Algebra. 2nd Edition. Arnold, London (et al.) 1991. MR1144518
• E. Artin: Galois theory . Harri Deutsch publishing house , Berlin (among others) 1968.
• PM Cohn: Algebra . Volume 2. 9th edition. John Wiley & Sons , London (et al.) 1989, ISBN 0-471-92234-X . MR1006872
• Richard Dedekind: On the theory of whole algebraic numbers . With a preface by B. van der Waerden. Friedr. Vieweg & Son , Braunschweig 1964. MR0175878
• Christian Karpfinger, Kurt Meyberg: Algebra . Groups - rings - bodies. Spektrum Akademischer Verlag , Heidelberg 2009, ISBN 978-3-8274-2018-3 .
• Kurt Meyberg: Algebra . Part 2 (=  mathematical basics for mathematicians, physicists and engineers ). Carl Hanser Verlag , Vienna 1976, ISBN 3-446-12172-2 . MR0460011
• BL van der Waerden: Algebra I . 9th edition. Springer Verlag , Berlin (among others) 1993, ISBN 3-540-56799-2 .

Footnotes and individual references

1. ↑ ^{a b} Meyberg: Algebra. Part 2. 1975, pp. 63-65
2. ^ Artin: Galois theory. 1968, pp. 28-30
3. ↑ This is where the fact that there is a commutative body comes into play . ${\displaystyle K}$
4. ↑ van der Waerden: Algebra I. 1993, pp. 159-163
5. ^ Karpfinger-Meyberg: Algebra. Groups - rings - bodies. 2009, p. 288
6. ↑ ^{a b} Cohn: Algebra vol. 2. 1989, pp. 81.84
7. ^ Allenby: Rings, Fields and Groups. 1991, p. 295
8. ↑ Cohn refers to p. 50 of the reprint of Dedekind's work On the Theory of Whole Algebraic Numbers , published in 1964 by Vieweg, Braunschweig . There the result appears as I. in § 166 and it literally means: If a group consists of different permutations of the body , and is the body of , then and the rest of is the identical permutation of . ${\displaystyle \Pi }$ ${\displaystyle n}$ ${\displaystyle \pi }$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle \Pi }$ ${\displaystyle (M,A)=n}$ ${\displaystyle \Pi }$ ${\displaystyle A}$
9. ^ Kurt Meyberg: Algebra , part 2. Carl Hanser Verlag, Vienna 1976, p. 73.
10. ^ Kurt Meyberg: Algebra , part 2. Carl Hanser Verlag, Vienna 1976, p. 75.