Alphabeta Math

Algebraic Extensions, Extension Degree, and Finite Fields

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✓ 29 results · all verified · 15 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 14 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Linear Algebra
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

The degree [K:F]=dim_F K of a finite field extension

Proposition

A finite extension has degree one if and only if the two fields are equal

Lemma

Products of bases form a basis in a tower of finite extensions

Theorem

Tower law for finite extensions: [L:F]=[L:K][K:F]

Corollary

The degree of an intermediate field divides the degree of a finite extension

Theorem

Every finite field extension is algebraic

Corollary

An element is algebraic over F if and only if its simple extension F(a)/F is finite

Definition

Finitely generated field extensions F(a₁,…,aᵣ)

Theorem

An extension generated by finitely many algebraic elements is finite

Theorem

The elements of an extension algebraic over the base field form a subfield

Definition

The relative algebraic closure of F in an extension K

Theorem

Algebraicity is transitive in towers of field extensions

Proposition

The relative algebraic closure of F in K has no further algebraic elements inside K

Proposition

For finite subextensions in a common field, [EE':F]≤ [E:F][E':F]

Definition

The prime subfield as the intersection of all subfields

Theorem

The characteristic of a field is zero or a prime number

Theorem

A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p

Theorem

The binomial theorem over an arbitrary commutative ring

Lemma

A prime p divides C(p, k) for 0<k<p

Theorem

Frobenius x↦ xᵖ is an injective endomorphism in characteristic p, and an automorphism for finite fields

Definition

Finite fields and their order

Theorem

Every finite field has order pⁿ for a unique prime characteristic p and positive integer n

Theorem

The multiplicative group F_q^× of a finite field is cyclic

Lemma

In characteristic p, the roots of x^pⁿ-x form a subfield and are all simple

Theorem

For every prime p and n≥1, a field with pⁿ elements exists

Proposition

A field with q elements is the splitting field of x^q-x over its prime subfield

Theorem

Finite fields of the same order are isomorphic

Theorem

The subfields of F_pⁿ are the unique fields F_pᵈ for positive divisors d of n

Theorem

Over F_q, x^qⁿ-x is the product of all monic irreducibles whose degrees divide n

Corollary

For every finite field F_q and every n≥1, a monic irreducible polynomial of degree n exists

Corollary

Every finite extension of a finite field is simple

Definition

Algebraically constructible real numbers as the smallest real subfield closed under positive square roots

Theorem

A real number is algebraically constructible exactly when it lies in a finite tower of real quadratic adjunctions

Corollary

An algebraically constructible real algebraic number has degree over ℚ equal to a power of two

Counterexample

The real cube root of two is algebraic but not algebraically constructible

Sources