Alphabeta Math

Splitting Fields

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✓ 16 results · all verified · 11 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 5 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Polynomials that split and splitting fields of a polynomial or a family of polynomials

Lemma

Kronecker's one-root step: adjoining a root removes a linear factor and lowers the remaining degree

Theorem

Every nonzero polynomial over a field has a splitting field

Corollary

Every finite family of nonzero polynomials has a splitting field, obtained from their product

Corollary

Inside a common extension, the splitting field of fg is the composite of the splitting fields of f and g

Lemma

A field isomorphism transports polynomials coefficientwise and carries roots, factorizations, and splitting to roots, factorizations, and splitting

Lemma

A base-field isomorphism extends across simple adjunctions of corresponding roots of an irreducible polynomial

Theorem

A base-field isomorphism extends to an isomorphism between splitting fields of corresponding polynomials

Corollary

Any two splitting fields of a polynomial are isomorphic over the base field

Theorem

A degree-n polynomial has a splitting field spanned over F by at most n! explicit root monomials

Proposition

After adjoining one nonzero root α of xⁿ-a, all roots are ζα with ζⁿ=1

Proposition

Every F-endomorphism of a splitting field permutes the distinct roots and is an automorphism

Definition

A normal algebraic extension is one in which every minimal polynomial with a root in the extension splits there

Proposition

If K/F is normal and F⊆ E⊆ K, then K/E is normal

Proposition

A nonempty intersection of normal subextensions inside a common algebraic extension is normal

Proposition

An algebraic extension that is a splitting field of a polynomial is normal

Theorem

An algebraic extension generated by elements whose minimal polynomials split in it is normal

Proposition

A normal extension generated by finitely many elements is the splitting field of the product of their minimal polynomials

Sources