Thread: Application of "Tower Theorem"

I can deduce from the "Tower theorem" (for extensions of fields) that if [K:F] is prime then there is no field E such that . I can't see how this relates to the last part of the question - how do we go about it?

"Assuming that fields of order 32 exist, show that there are exactly 6 irreducible polynomials of degree 5 in the ring "

Originally Posted by Amanda1990

I can deduce from the "Tower theorem" (for extensions of fields) that if [K:F] is prime then there is no field E such that . I can't see how this relates to the last part of the question - how do we go about it?

Say that then . However, is prime which forces therefore . Hence, there is no .

"Assuming that fields of order 32 exist, show that there are exactly 6 irreducible polynomials of degree 5 in the ring "

Let be a field of order , then must contain a subfield which is isomorphic to . Notice that . If then , but since is prime it follows by above that . Therefore, if is the mimimal polynomial for then must be an irreducible monic polynomial of degree in . There are, elements in and for each one the mimimal polynomial has roots in , therefore it means there has to be elements in that give rise to each different polynomial of degree .