Dedekind's Prague phrase

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The Prague sentence of Richard Dedekind is a set of number theory and algebra on polynomials prospectus, which is Dedekind's ideal theory at the beginning of abstraction. It can be shown that it applies precisely in completely closed rings and can therefore be derived for normalized polynomials with the Gaussian levitation lemma (applicable if the four-number theorem applies in the ring ).

are

${\displaystyle b(X)=\sum _{i=1}^{m}{b_{i}X^{i}}}$and polynomials of the polynomial ring of an integrity ring with a quotient field ${\displaystyle c(X)=\sum _{j=1}^{n}{c_{j}X^{j}}}$ ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle K}$

and is for one , so is for all . ${\displaystyle a|b(X)c(X)}$ ${\displaystyle a\in R}$ ${\displaystyle a|b_{i}c_{j}}$ ${\displaystyle 0\leq i\leq m,0\leq j\leq n}$

The following statements are synonymous with this sentence:

1. is completely completed. ${\displaystyle R}$

2. In , the Gaussian levitation lemma applies to normalized polynomials ${\displaystyle R[X]}$

If normalized polynomials are with , then are . ${\displaystyle g,h\in K[X]}$ ${\displaystyle gh=f\in R[X]}$ ${\displaystyle g,h\in R[X]}$

In other words, the theorem says: Let f, g be polynomials in a variable whose coefficients are algebraic numbers. If all the coefficients of the product are algebraic integers, the product of every coefficient of f by every coefficient of g is also an algebraic integer. ${\displaystyle f\cdot g}$

Dedekind published the theorem in 1892 as a generalization of a theorem by Carl Friedrich Gauß (Lemma von Gauß, Disquisitiones Arithmeticae , Article 42): The polynomials f, g in a variable have rational coefficients. If the coefficients of f, g are not all integers, then the coefficients of not all can not be integers either. The term Prague Sentence comes from the fact that Dedekind published it in the communications of the German Mathematical Society in Prague. ${\displaystyle f\cdot g}$

Dedekind was not aware that Leopold Kronecker had published the sentence ten years earlier, but in a very brief form and in an obscure (idiosyncratic) formulation (Harold Edwards).

Even if Hilbert in particular gave preference to the (similar) Hurwitzian structure of the ideal theory in his number report (Dedekind's Prague theorem serves to prove that for every ideal in a maximum order of a number field there is an ideal such that it is a main ideal), Dedekind's Prague theorem plays practically no role in today's algebra and number theory, since one prefers (in Dedekind's sense via E. Noether to Grothendieck ) conceptual proofs with abstract principles such as complete isolation. ${\displaystyle a\neq (0)}$ ${\displaystyle b\neq (0)}$ ${\displaystyle ab}$

literature

• Harold Edwards : Divisor Theory , Springer 1990
• Franz Lemmermeyer : On the number theory of the Greeks , Part 2, Mathematical Semester Reports, Volume 55, 2008, pp. 181–195

Individual evidence

1. ^ ^{A b} Edwards, Divisor Theory, 1990, p. 2
2. ↑ Dedekind, About an arithmetic theorem by Gauß, Mitteilungen der Deutschen Mathematischen Gesellschaft Prag, 1892, pp. 1–11, Dedekind, Werke, Volume 2, pp. 28–38
3. ↑ Kronecker, On the theory of forms of higher levels, monthly reports of the Academy of Sciences in Berlin, 1883, pp. 957–960, Kronecker, Werke, Volume 2, pp. 419–424
4. ^ David Hilbert: The theory of algebraic number fields, annual report of the German Mathematicians Association, V.4 p.175-546 1897