### Gille-Szamuely

Posted on May 25, 2019Here are my exercise solutions and notes for Gille and Szamuely’s *Central Simple Algebras and Galois Cohomology*.

**1.1.** Since the map is linear, for we must have To have we must have Computing the product and taking , we find that only if Setting and we see that we must also have Finally, setting and we see that Thus

**1.3.** Let Recall that for some by Lemma Therefore let us pick the basis for and observe that is a basis for Thus the quaternion norm agrees with the relative field norm for elements of

**1.5.** Recall that splits over if and only if the curve defined by has a -rational point. Observe that no -rational point on has -coordinate zero since such a point would have for both , a contradiction. So all -rational points lie on the affine curve By a classic result from the Gaussian integers, a prime may be written as the sum of two squares if and only if (see the first section of the first chapter of Neukirch’s algebraic number theory book, for example).

**Lemma.** If such that then or

Let Computing we find that Therefore if and only if or

**1.6.** Let be an isomorphism. Take such that , and similarly such that Then let and

Define Observe that is in fact a subpsace of of dimension and is in fact the span of (assuming that we see that implies that ).

Since , by the lemma above we have Therefore we have -dimensional subspaces and in Thus there is a nonzero element in their intersection. Since we see that Similarly, as we find

Therefore is an alternate basis for and is an alternate basis for Taking we therefore have and