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Every finite field extension is algebraic
Statement
Every finite field extension is algebraic: each is a root of a nonzero polynomial in .
Facts & Assumptions
Given: A finite extension of degree and an element .
Degree means that has an -basis of size (The degree of a finite field extension).
Any vectors in a space spanned by vectors are linearly dependent (If has a spanning set with elements, then every linearly independent subset of is finite with at most elements; in particular has no linearly independent subset equinumerous with ).
An element is algebraic over when a nonzero polynomial in vanishes at it (Algebraic and transcendental elements and algebraic extensions).
Proof
The vectors lie in the -dimensional -space , so [L2] gives coefficients , not all zero, with .
The polynomial is nonzero and satisfies , so is algebraic by [L3].
Since was arbitrary, the extension is algebraic. The case cannot occur for a field extension because .
Depends on
- The degree $[K:F]=\dim_F K$ of a finite field extension
- Algebraic and transcendental elements and algebraic extensions
- If $V$ has a spanning set with $n$ elements, then every linearly independent subset of $V$ is finite with at most $n$ elements; in particular $V$ has no linearly independent subset equinumerous with $\mathbb{N}$
Used by
- An element is algebraic over F if and only if its simple extension F(a)/F is finite Corollary
- Every finite extension of a perfect field is simple Corollary
- The algebraic numbers in ℂ form an algebraic closure of ℚ Corollary
- An algebraic extension need not be finite Counterexample
- A finite normal extension is separable over its purely inseparable fixed field Lemma
- A finite-type field has finite relative algebraic constants Lemma
- Conjugates of an average of roots of unity Lemma
- Finite purely inseparable rational extensions admit a finite Frobenius envelope Lemma
- Finite-type field extensions with zero Ω Lemma
- A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension Proposition
- For finite subextensions in a common field, [EE':F]≤ [E:F][E':F] Proposition
- The relative algebraic closure of F in K has no further algebraic elements inside K Proposition
- Algebraicity is transitive in towers of field extensions Theorem
- Assuming Choice, every field has an algebraic extension containing roots of all nonconstant base polynomials Theorem
- Polynomial algebras over fields have finite integral closures Theorem
- The elements of an extension algebraic over the base field form a subfield Theorem
Dependency tree · two levels
17 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
- A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 1 (standard reference, not scraped)