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

Depends on

• Field extensions, generated subrings $F[S]$, generated subfields $F(S)$, and simple extensions
• Evaluation and roots of a polynomial in a commutative target ring
• Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree

Used by

• A characteristic polynomial that splits into distinct linear factors forces diagonalisability Corollary
• Every finite family of nonzero polynomials has a splitting field, obtained from their product Corollary
• If the minimal polynomial splits, V is the direct sum of the stabilised generalised eigenspaces Corollary
• Inside a common extension, the splitting field of fg is the composite of the splitting fields of f and g Corollary
• Vieta's formulas identify the coefficients of a split monic polynomial with elementary symmetric functions of its roots Corollary
• A normal algebraic extension is one in which every minimal polynomial with a root in the extension splits there Definition
• A polynomial is solvable by radicals when its splitting field lies in a radical extension Definition
• The cyclotomic extension K(μₙ) as a splitting field of tⁿ-1 Definition
• The Galois group of a separable polynomial Definition
• The resolvent cubic of a monic quartic Definition
• Over F₂, x⁴+x²+1=(x²+x+1)² has two distinct roots, each repeated, in its four-element splitting field Example
• The base field is the splitting field of the empty family and of every nonzero constant polynomial Example
• The splitting field of {x²-2,x²-3} over ℚ is ℚ(√2,√3) Example
• The splitting field of x²-2 over ℚ is ℚ(√2), with roots ±√2 Example
• The splitting field of x⁴+2x²-8 over ℚ is ℚ(√2,i) Example
• x⁵-2 over ℚ is solvable by radicals although it is a quintic Example
• x⁵-6x+3 over ℚ is not solvable by radicals Example
• FALSE: If the minimal polynomial splits, then the endomorphism is diagonalisable False statement
• A field isomorphism transports polynomials coefficientwise and carries roots, factorizations, and splitting to roots, factorizations, and splitting Lemma
• A finite normal extension is separable over its purely inseparable fixed field Lemma
• A proper rational function with split denominator has a unique repeated-pole partial-fraction expansion Lemma
• Kronecker's one-root step: adjoining a root removes a linear factor and lowers the remaining degree Lemma
• To prove the fundamental theorem of algebra, it suffices to split every real polynomial over ℂ Lemma
• A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension Proposition
• A field with q elements is the splitting field of x^q-x over its prime subfield Proposition
• A normal extension generated by finitely many elements is the splitting field of the product of their minimal polynomials Proposition
• After adjoining one nonzero root α of xⁿ-a, all roots are ζα with ζⁿ=1 Proposition
• An algebraic extension that is a splitting field of a polynomial is normal Proposition
• Every F-endomorphism of a splitting field permutes the distinct roots and is an automorphism Proposition
• A base-field isomorphism extends to an isomorphism between splitting fields of corresponding polynomials Theorem
• A complex polynomial of degree n has exactly n roots counted with multiplicity Theorem
• A degree-n polynomial has a splitting field spanned over F by at most n! explicit root monomials Theorem
• A monic irreducible of degree d over F_q has the d distinct roots α,α^q,…,α^qᵈ⁻¹ Theorem
• A polynomial Galois group acts faithfully on its roots Theorem
• An algebraic extension generated by elements whose minimal polynomials split in it is normal Theorem
• An endomorphism is diagonalisable if and only if its characteristic polynomial splits and every eigenvalue's geometric multiplicity equals its algebraic multiplicity Theorem
• Equivalent characterizations of a finite Galois extension Theorem
• Every nonzero polynomial over a field has a splitting field Theorem
• For every prime p and n≥1, a field with pⁿ elements exists Theorem
• For monic f,g of degrees n,m splitting in a common extension, Res(f,g)=∏ᵢ₌₁ⁿ∏ⱼ₌₁ᵐ(αᵢ-βⱼ) Theorem

…and 9 more results.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• T. Judson, Abstract Algebra: Theory and Applications, Section 21.2 (standard reference, not scraped)
• J. S. Milne, Fields and Galois Theory, Chapter 2 (standard reference, not scraped)