Gille-SzamuelyPosted on May 25, 2019
Here 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