Unformatted text preview: Math 4108 midterm 2, Spring 2010
April 30, 2010
1. Deﬁne the following terms a. Galois group of a ﬁeld K over a ﬁeld F (Also, what properties must K have in order for the Galois group to even exist?) b. Lower Central Series. c. Fundamental Theorem of Finitely Generated Modules over a PID. d. normal ﬁeld extension. e. nilpotent group. 2. Compute the derived series for S4 . Explain your work. 3. Compute the Galois group for the polynomial x4 − 2. What basic group is it isomorphic to? 4. Suppose that α, β ∈ Fpn . Let Hα and Hβ be the multiplicative subgroups generated by the powers of α and β , respectively. Prove that |Hα | = |Hβ | =⇒ Hα = Hβ . 5. Prove that the length of the Upper Central Series for a nilpotent group is always at least as long as the Derived Series for the group (Use whatever facts about Upper and Lower series you need – if you are unsure of what you are allowed to use, just ask me.) 1 ...
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