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[parent] a finite extension of fields is an algebraic extension (Theorem)
Theorem 1   Let $ L/K$ be a finite field extension. Then $ L/K$ is an algebraic extension.
Proof. In order to prove that $ L/K$ is an algebraic extension, we need to show that any element $ \alpha\in L$ is algebraic, i.e., there exists a non-zero polynomial $ p(x)\in K[x]$ such that $ p(\alpha)=0$.

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

$\displaystyle [L\colon K]=n$
for some $ n\in \mathbb{N}$.

Consider the following set of “vectors” in $ L$:

$\displaystyle \mathcal{S}=\{ 1, \alpha, \alpha^2,\alpha^3,\ldots,\alpha^n\}$
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 $ k_i\in K,\ 0\leq i \leq n$, not all zero, such that
$\displaystyle k_0+k_1\alpha+k_2\alpha^2+k_3\alpha^3+\ldots+k_n\alpha^n=0$
Thus, if we define
$\displaystyle p(X)=k_0+k_1X+k_2X^2+k_3X^3+\ldots+k_nX^n$
then $ p(X)\in K[X]$ and $ p(\alpha)=0$, as desired.

$ \qedsymbol$

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



"a finite extension of fields is an algebraic extension" is owned by alozano.
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See Also: algebraic, algebraic extension, proof of transcendental root theorem

Keywords:  algebraic, polynomial, finite

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the field extension $\mathbb{R}/\mathbb{Q}$ is not finite (Corollary) by alozano
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Cross-references: converse, linearly dependent, cardinality, dimension, vector space, finite dimensional, fields, polynomial, algebraic, order, algebraic extension, finite field extension
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This is version 3 of a finite extension of fields is an algebraic extension, born on 2003-09-11, modified 2003-09-17.
Object id is 4725, canonical name is AFiniteExtensionOfFieldsIsAnAlgebraicExtension.
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AMS MSC: 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions)
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