a finite extension of fields is an algebraic extension

Theorem 1.

Let L/K be a finite field extension. Then L/K is an algebraic extensionMathworldPlanetmath.


In order to prove that L/K is an algebraic extension, we need to show that any element αL is algebraic, i.e., there exists a non-zero polynomialPlanetmathPlanetmath p(x)K[x] such that p(α)=0.

Recall that L/K is a finite extension of fields, by definition, it means that L is a finite dimensional vector spaceMathworldPlanetmath over K. Let the dimensionPlanetmathPlanetmath be


for some n.

Consider the following set of “vectors” in L:


Note that the cardinality of S is n+1, one more than the dimension of the vector space. Therefore, the elements of S must be linearly dependent over K, otherwise the dimension of S would be greater than n. Hence, there exist kiK, 0in, not all zero, such that


Thus, if we define


then p(X)K[X] and p(α)=0, as desired.

NOTE: The converse is not true. See the entry “algebraic extension” for details.

Title a finite extension of fields is an algebraic extension
Canonical name AFiniteExtensionOfFieldsIsAnAlgebraicExtension
Date of creation 2013-03-22 13:57:30
Last modified on 2013-03-22 13:57:30
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Theorem
Classification msc 12F05
Related topic Algebraic
Related topic AlgebraicExtension
Related topic ProofOfTranscendentalRootTheorem