Kronecker's Theorem (body theory)

The set of Kronecker ( English : Kronecker's theorem ) of the field theory is one of the tenets of the mathematician Leopold Kronecker , which within the Algebra are located. The theorem deals with the question of the existence of zeros of polynomials over commutative fields and as such is fundamental in the theory of decay fields .

Formulation of the sentence

The sentence can be summarized as follows:

(1) For any irreducible polynomial over a commutative field , a finite field extension can always be found in which has a zero and the degree of extension corresponds to the degree of the polynomial ; ${\displaystyle p\in K[X]}$ ${\displaystyle K}$ ${\displaystyle L/K}$ ${\displaystyle p}$

so in such a way that always the equations

(1a) for at least one ${\displaystyle p(x_{0})=0}$ ${\displaystyle x_{0}\in L}$

(1b) ${\displaystyle [L:K]=\deg(p)}$

are fulfilled.

(2) For every non-constant polynomial over a commutative field there is always a finite field extension in which has a zero and whose degree of extension satisfies the inequality with regard to the degree of the polynomial . ${\displaystyle f\in K[X]}$ ${\displaystyle K}$ ${\displaystyle L/K}$ ${\displaystyle f}$ ${\displaystyle [L:K]\leq \deg(f)}$

Inference

The Kronecker theorem has the following result:

For every commutative field and every polynomial there is a decay field , for whose degree of expansion the inequality exists in relation to the degree of the polynomial . ${\displaystyle K}$ ${\displaystyle f\in K[X]}$ ${\displaystyle S_{f}/K}$ ${\displaystyle [S_{f}:K]\leq \deg(f)!}$

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
• 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 .

References and footnotes

1. ^ Allenby: Rings, Fields and Groups. 1991, p. 140 ff
2. ^ Artin: Galois theory. 1968, p. 24 ff
3. ^ Cohn: Algebra vol. 2. 1989, p. 69 ff
4. ^ Meyberg: Algebra. Part 2. 1975, p. 28 ff
5. ↑ The exclamation mark stands for the factorial function .
6. ↑ As can be seen, the disintegration body is clearly determined except for isomorphism . ${\displaystyle S_{f}}$