As preparation for my upcoming PhD written preliminary exam, the following are some sample questions in Algebra, Representation theory, Category Theory and Homological Algebra from various sources (thanks to this mathstackexchange post):
- Let be the field and algebraically closed field containing . Fix , the polynomial . Let be the field of rational functions in an indeterminate .
Prove is a 2-dimensional vector space over . (Checking vector space condition is easy; Dimension being 2 follows from .)
Let be the extension of obtained by adjoining a root of . Show that is a Galois extension with Galois group isomorphic to . (To check is Galois over , note that for , so contains all the roots of ; A bijection from to is given by where . The picture is that is the function field of an abelian Galois cover of of degree .)
How many fields are there such that other than and ? (By Galois correspondence the intermediate fields correspond to subgroups of order , so there are many.)
- Let be a finite group, a normal subgroup of , be a Sylow -subgroup of .
(Frattini’s argument) Show that where is the normalizer subgroup of in . (Given , since is Sylow -subgroup of , there exists such that .)
Show that if every maximal proper subgroup of is normal in , then so is any Sylow -subgroup of . (If is not normal, then we can let be a maximal proper subgroup containing . Then Frattini’s argument implies that , contradiction.)
- Let be a finite group, irreducible characters of , subgroup of , irreducible character of . Show that the integers for which satisfy (By Frobenius reciprocity, ; Since is irreducible, this means that for some other . Thus .)
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