Bachelor’s Project: Realising Frobenius groups as Galois groups (2022)

Advisor: Ian Kiming.

Abstract: \(F_{pl}\), as Galois groups. The first half of the project focuses on realising the general case. At the end of this first half, a class of examples for the case \(l=\frac{p-1}{2}\) is constructed. In the proof of the general case, we have to make a certain assumption. Hence, we will, in the second half, consider a special case, where we can confirm that the assumption holds. This special case is for \(l=2\), i.e., \(F_{2p}=D_p\). The main theorem of this second half is that given a quadratic extension, \(D_p\) can be realised in infinitely many different ways. To do this we will need to introduce notions from class field theory. Amongst other things, we will need a formula for the class number, which will be introduced and proven.

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