Question

If $\mathit{K}$is a finite field show that $\mathit{K}$is an algebraic extension of $\mathit{F}$.

Short answer

$K$is an algebraic extension of$F$.

Step-by-step solution

Definition of extension.

Let $K$ an extension field of $F$. Then $K$ is said to be algebraic extension of $F$ if every element of $K$is algebraic over$F$.

Step 2:Showing that K is algebraic extension of F .

Let $E$be an extension field of $F$. $E$is finite extension of $F$if $E$ is of finite demision$n$ as a vector space over $F$. Also $\left[E:F\right]=n$.

Then by definition given above

$\left[K:F\right]=n$

Let . $u\in K$The set of n+1 elements$\left\{1,u,u^2.....u^{n+1}\right\}$ is linearly dependent.

Hence there are $a_i\in F$ not all zero such that

$a_0+a_{1}u+a_2{u}^2+.......a_n{u}^{n+1}=0$

Which implies that $u$ is algebraic over $F$.

Since $u$ is arbitrary. Therefore the finite extension field $K$ is an algebraic extension of $F$.