Dummit Foote Exercise 13.3.15 A Field F is said to be formally real if $-1$ can not be written as sum of squares in F.
let $f(x)$ be an irreducible polynomial in $F[x]$ of odd degree with $\alpha$ as a root. 
Now the Question is Prove that $F(\alpha)$ is also Formally real.
What i have done so far is to check that i can not exclude the condition of degree of $f(x)$ being odd. If $f(x)$ is of even order then this need not be the case. For example for  $f(x)=x^2+1$ over $\mathbb{Q}$ we have $i$ as a root and in $\mathbb{Q}(i)$ we have $-1=i^2$. So, i have to use that $f(x)$ is a polynomial of odd degree. 
But i am unable to decide how to proceed with this.
Before this exercise there is an exercise if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$. I have proved this already. I somehow feel i should use this result to prove given result. 
But no progress in this way too..
Any small hint would be appreciated.
Thank You.
 A: Have you read the hint given in the book? For example suppose there is an irreducible polynomial $f$ of odd degree such that $F(\alpha)$ is not formally real, $\alpha$ a root of $f$. Choose such an irreducible polynomial of minimal degree. If $-1$ can be written as a sum of squares $p_1(\alpha)^2,\ldots,p_m(\alpha)^2$ then this means we have an equation
$$-1 = p_1(\alpha)^2 + \ldots + p_m(\alpha)^2.$$
So this means that under the isomorphism $F(\alpha) \cong F[x]/(f(x))$, the element $\alpha$ on the L.H.S corresponds to $\overline{x}$ the residue of $x$ in the quotient. So this means in the quotient ring we have
$$-1 = p_1(\overline{x})^2 + \ldots + p_m(\overline{x})^2$$
and so therefore in the polynomial ring $F[x]$ we have an equation $-1 + f(x)g(x) = p_1(x)^2 + \ldots + p_m(x)^2$. So by comparing degrees, we see that $\deg g$ is odd and furthermore $\deg g < \deg f$.


Question 1: Why is $\deg g < \deg f$? Hint: What is the maximal degree that a "polynomial"  $p(\overline{x})$ in the quotient can have?
Question 2: I have spelt out enough details, can you use the hint in Dummit and Foote to complete the rest?


