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Every -endomorphism of a splitting field permutes the distinct roots and is an automorphism
Statement
Let be a splitting field of a nonzero polynomial . Every field homomorphism that fixes maps the finite set of distinct roots of bijectively to itself. Consequently is surjective and hence is an -automorphism of .
Facts & Assumptions
Given: A splitting field of and an -endomorphism .
A unital homomorphism between fields is injective (Field homomorphism and embedding).
A field isomorphism carries roots of a polynomial to roots of the transported polynomial (A field isomorphism transports polynomials coefficientwise and carries roots, factorizations, and splitting to roots, factorizations, and splitting).
A nonzero degree- polynomial over a domain has at most distinct roots (A nonzero polynomial of degree over an integral domain has at most distinct roots).
An injection from a finite set to itself is a bijection (A subset of a finite set is finite, with , and equality holds if and only if ).
A splitting field is generated over the base field by the roots of its polynomial (Polynomials that split and splitting fields of a polynomial or a family of polynomials).
Proof
By [F1], is injective and hence an isomorphism from onto its image. Since it fixes the coefficients of , direct evaluation gives for every root of .
The distinct-root set is finite by [F3], and step 1.1 restricts to an injection . By [F4], this restriction is a bijection, so permutes the roots. This remains true when is empty.
The image contains and every root of by step 2.1. Since those elements generate by [F5], one has . Thus is surjective and is an -automorphism.
Depends on
Polynomials that split and splitting fields of a polynomial or a family of polynomials
A field isomorphism transports polynomials coefficientwise and carries roots, factorizations, and splitting to roots, factorizations, and splitting
Field homomorphism and embedding
A nonzero polynomial of degree over an integral domain has at most distinct roots
A subset of a finite set is finite, with , and equality holds if and only if