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I'm having some trouble solving this homework problem and understanding what the hint is trying to tell me.

Suppose $K$ is a field extension of $F$ of finite degree. Prove that if $\alpha$ is in $K$, then there is an irreducible polynomial $f(x)$ in $F[x]$ having $a$ as a root. (Hint: If $[K : F]=n$, consider $1,a,a^2,...,a^n$)

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Here's a little hint: the family of $1,a,a^2,\dots$ generates a subring of $K$ which is also a $F$-vector space, so subspace of $K$ as $F$-vector space.

From this you could easily conclude using a little of linear algebra.

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The nastiest point of the question is the insistence that $f$ should be irreducible. However one can easily see that the existence of a nonzero polynomial $p$ such that $p[\alpha]=0$ suffices: decompose $p$ into (a nonzero constant and) irreducible factors $f_i$; then the given condition $p[\alpha]=0$ means that $\prod_if_i[\alpha]=0$, and from the fact that $K$ is an integral domain (as all fields are), it follows at least one of the $f_i[\alpha]$ must be zero, and then we can take our irreducible$~f$ to be $f_i$.

Now if such a nonzero polynomial$~f$ would not exist, this would mean that the $F$-linear map $F[x]\to K$ substituting $\alpha$ for$~x$, that is the map $p\mapsto p[\alpha]$, is injective (it sends no nonzero polynomial to$~0$). Then its image would an infinite-dimensional $F$-vector subspace of$~L$, but since $L$ is finite dimensional as $F$-vector space, this cannot be.

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