Mathematics 352 – Abstract Algebra II Midterm Exam 1 – February 27, 2008

Directions: Place all work you want to have considered for credit in the blue exam book. You have until 9:00pm to work on the exam if you need that much time. There are 100 regular and 10 Extra Credit points. Some partial credit will be given for relevant definitions, statements of theorems, etc. Even if you cannot see how to complete some part, don’t leave it blank.

I. Terminology.

A) (5) Let G be a group and let x in G. What is the conjugacy class of x in G?

B) (5) Let K be an extension field of the field F and let u ∈ K. What does it mean to say that u is transcendental over F?

C) (5) Let K be an extension field of the field F. What does it mean to say that K is normal over F?

II. A) (10) State the Structure Theorem for finite abelian groups.

B) (10) Let G = Z_{42} ⊕ Z_{36} ⊕ Z_{84}. What are the elementary divisors and invariant factors of G?

C) (5) What is the Sylow 7-subgroup of G from part B? Is this a cyclic group?

III.

A) (20) State and prove the First Sylow Theorem for finite groups.

B) (10) Use the Sylow Theorems to show that every group of order 33 is cyclic.

IV. A) (5) Let K be an extension field of F, and let u ∈ K. Show that if u^{2} is algebraic over

F, then u is also algebraic over F.

B) (10) Show that Q(\sqrt{2}, \sqrt{3}) is a splitting field of f(x) = x^{4} + 2x^{3} − 8x^{2} − 6x − 1 over Q. (Hint: Look for quadratic factors of f(x).)

V. (15) Show that if F ⊆ E ⊆ K are field extensions and [E : F], [K : E] are both finite, then [K : F] = [K : E][E : F].

Extra Credit (10) Let p(x) and q(x) be irreducible polynomials in F [x] such that deg p(x) and deg q(x) are relatively prime integers. Show that if u is a root of p(x) in some extension field of F, then q(x) is also irreducible in F (u)[x].