Exercise 6, § 19

Let H be the quaternion field of type $(-1,0,-1)$ over $\mathbf{Q}$. a) An extension K of $\mathbf{Q}$ splits H if and only if $-1$ is the sum of two squares in K (apply Propositions 3 of VIII, p. 363 and 4 of VIII, p. 364).

b) Let K be a cyclic extension of $\mathbf{Q}$ of degree $2^n$ that splits H. Prove that no proper subfield of K splits H (observe that the unique subfield of K of degree $2^{n-1}$ is the intersection of K and a maximal ordered field containing $\mathbf{Q}$).

c) Let $k$ be an integer $\geqslant 1$ and $p$ a prime divisor of $2^{2^k}+ 1$. Prove that the greatest integer $n$ such that $2^n$ divides $p-1$ is $> k$ (observe that we have $2^{p-1}\equiv 1$ modulo $p$). Let Ω be a primitive $p$-the root of 1 and $E =\mathbf{Q}(\Omega )$. Prove that $-1$ is the sum

of two squares in E (let F be the extension E( $-1$); show that $-1$ belongs to $N_{F/E}(F^*)$ by proving that this is true for $\omega$ and $1 +\omega^h$ and by using the identity $\prod^{r-1}_{j=0}(1 + T^{2^j}) =\sum^{2^r-1}_{h=0}T^h$).

d) Prove that a cyclic extension of $\mathbf{Q}$ of degree $2^n$ contained in E splits H (apply VIII, p. 283, Corollary 2).