# Finite Field Extensions

## Homework Statement

Let F be a field, and suppose that alpha is algebraic over F. Prove that if [F(alpha):F] is odd, then F(a^2)=F(a). {For those unfamiliar with notation [] denotes degree of extension and F(alpha) means F adjoined with alpha.)

## The Attempt at a Solution

Since [F(alpha):F] is odd the basis of F(alpha) consists of an odd number of elements. We also know that b in F(alpha) has the form c_0+c_1alpha+c_2alpha^2+...+c_n-1alpha^(n-1). I really have no idea how to proceed from here. Help would be much appreciated. Thanks.

First of all, have you learned the Tower Law yet (i.e., if $$L \subseteq K \subseteq M$$ are fields, then $$[M : L] = [M : K] [K : L]$$)?

If you understand this, then the problem is relatively easy. It should be pretty obvious that $$F(\alpha^2) \subseteq F(\alpha)$$. Furthermore, $$F(\alpha^2) = F(\alpha)$$ if and only if $$[F(\alpha) : F(\alpha^2) ] = 1$$. Why is it true that if $$[F(\alpha) : F(\alpha^2)] \neq 1$$, then $$[F(\alpha) : F(\alpha^2)] = 2$$? Using the tower law, why would the latter contradict your assumptions about $$[F(\alpha):F]$$?

Thanks for the help, but sorry we haven't learned about the "tower law." Can you help me with the part where you said that [F(a):F(a^2)]=2 if not 1.

Well, for any field $$K$$ and any $$\beta$$, the degree of $$K(\beta)$$ over $$K$$ is the degree of the minimal polynomial of $$\beta$$ over $$K$$. We know that $$\alpha$$ satisfies the polynomial $$f(x) = x^2 - \alpha^2$$ with coefficients in $$F(\alpha^2)$$; hence, the minimal polynomial of $$\alpha$$ over $$F(\alpha^2)$$ must divide $$f(x)$$. Since the degree of $$f(x)$$ is 2, this implies that the degree of $$\alpha$$ over $$F(\alpha^2)$$ is less than or equal to 2.

The tower law is the following: For any fields $$L \subseteq K \subseteq M$$, we have $$[M:L] = [M:K] [K:L]$$. (I'm a little surprised that you were assigned this question if you haven't covered this, to be honest; maybe they want you to derive a special case of the tower law yourself?)

matt grime