Extensions of fields and dimension

Question

The following is a homework question:

Let M: N be a field extension, with a ∈ M algebraic over N. Show every element of N(a) is algebraic over N.

Can anyone give me a strategy to approach this?

Answer 1

If an element $\alpha$ is algebraic over a field $F$, then it is a root of an irreducible polynomial $f(x) \in F[x]$. Let's say that polynomial has degree $n$.

It follows that $F[\alpha]$ is a degree $n$ field extension over $F$. Now think about how "degree" is defined in terms of field extensions as vector spaces over the base field: if $[F[\alpha]:F] = n$, then $\{1, \beta, \beta^2, ..., \beta^{n} \}$ cannot be a linearly independent set for any $\beta \in F[\alpha]$. This means, for any $\beta \in F[\alpha]$, we can write: $$\sum_{k=0}^{n} c_k\beta^k = 0\ \ \ \text{ for } \ c_k \in F \text{ not all zero}$$

How can you use this information to construct a polynomial for which $\beta$ is a root?