field extension with Galois group
Let . We will show that is Galois over and that (the group of quaternions).
We begin by showing that . Let . Claim that . To show that they are not equal, we show that , i.e. that is not a square in . If it were, say , take to be the element
Then , so
But , and thus , so , a contradiction. Thus . We show that by showing that .
So and thus . Then .
so that is a degree polynomial with as a root. In fact,
Furthermore, it is easy to see that each of these roots lies in , for
so dividing through by we see that
acts transitively on the roots of , and , so an element of is determined by the image of . Thus the elements of are the automorphisms of that map to any of the eight roots of . Let
and let be elements of .
, so . This is an equation in , so regarding as an automorphism of , it must be the automorphism . Since , we have and thus that . It follows that is an element of order (http://planetmath.org/OrderGroup) in .
Similarly, , so , so that , regarded as an automorphism of , must be . Since , we have , so that , and is also an element of order (http://planetmath.org/OrderGroup) in . Note also that , so that .
Looking at ,
and thus . So .
|Title||field extension with Galois group|
|Date of creation||2013-03-22 17:44:28|
|Last modified on||2013-03-22 17:44:28|
|Last modified by||rm50 (10146)|