4108_midterm2 - Math 4108 midterm 2, Spring 2010 April 30,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online