Degree of Element of Finite Field Extension divides Degree of Extension

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Theorem

Let $F$ be a field whose zero is $0$ and whose unity is $1$.

Let $K / F$ be a finite field extension of degree $n$.

Let $\alpha \in K$ be algebraic over $F$.


Then the degree of $\alpha$ is a divisor of $n$.


Proof

Let $\alpha \in K$.

The dimension of $K / F$ considered as a vector space is $n$.


Let $S$ be the set defined as:

$S := \set {1, \alpha, \alpha^2, \ldots, \alpha_n}$

We have that:

$\card S = n + 1$

From Cardinality of Linearly Independent Set is No Greater than Dimension:

$S$ is linearly dependent on $F$.

Hence there are scalars $c_0, c_1, \ldots, c_n \in F$, not all zero, such that:

$c_0 + c_1 \alpha + c_2 \alpha^2 + \dotsb + c_n \alpha_n = 0$

which is a polynomial of degree $n$.

So $\alpha$ is algebraic over $F$ with degree no greater than $n$.


Consider $\map F \alpha$, the simple algebraic extension of $F$ on $\alpha$.

We have that:

$F \subseteq \map F \alpha \subseteq K$

Let $\alpha$ be algebraic over $F$ with degree $m$.

Then $\map F \alpha$ is a finite extension of $F$ with degree $\index {\map F \alpha} F = m$.

We also have that $K$ is a finite extension of $F$ with degree $n$.

So by Degree of Field Extensions is Multiplicative:

$\index K F = \index K {\map F \alpha} \index {\map F \alpha} F$

But $\index K F = n$ and $\index {\map F \alpha} F = m$

so it follows that:

$m \divides n$

The result follows.

$\blacksquare$


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