Mathematics 352 – Abstract Algebra II Final Exam – May 3, 2008

Directions:

Place all work you want to have considered for credit in the blue exam book. You have until 11:30am to work on the exam if you need that much time. There are 200 regular and 20 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. Answer any five of the following. You can earn up to 10 points Extra Credit for answering more than five. (A) (5) What is a Sylow p-subgroup of a finite group G? (B) (5) What is the class equation of a finite group? (C) (5) What does it mean to say that a point Q in the plane is constructible, starting from two

given points O = (0, 0), and P = (1, 0)? (D) (5) What is the degree of an extension field K of a field F (written [K : F])? (E) (5) What is the characteristic of a field? (F) (5) What is the Galois group, Gal_{F} K, of an extension K of a field F? (G) (5) What does it mean to say that an extension field K of a field F is Galois over F? (H) (5) What does it mean to say that a finite group G is solvable?

II. (A) (10) State the Structure Theorem for finite abelian groups. (B) (20) Classify all finite abelian groups of order n = 300. How many different isomorphism

classes are there? Give both the elementary divisors and the invariant factors for each isomorphism class.

III. (A) (15) State the three Sylow Theorems for finite groups. (B) (10) Show using the Sylow Theorems that every group G of order n = 99 has a non-trivial

normal subgroup N (that is, a normal subgroup N ≠ {e}, G). (C) (10 Extra Credit) Show that every group of order n = 99 is isomorphic either to ℤ_{9} × ℤ_{11} or

to ℤ_{3} × ℤ_{3} × ℤ_{11}.

IV. (10) Show that if K is an extension field of F and [K : F] < ∞, then every element of K is algebraic over F.

V. (15) It can be shown, using some slightly tedious but elementary manipulations with trig identities, that the number x = 2 cos (\frac{2π}{7} ) is a root of the equation x^{3} + x^{2} − 2x − 1 = 0. What does this imply about the constructibility of the regular 7-gon? Explain. VI. Let K be the splitting field of x^{64} − x ∈ ℤ_{2}[x], an extension field of F = ℤ_{2}. (A) (10) Show that the mapping φ : K → K given by φ(x) = x^{2} is an F -automorphism of K. (B) (10) Show that the mapping φ from part (A) is an element of order 6 in the Galois group

Gal_{F} K. (In fact, Gal_{F} K = ⟨φ⟩ is cyclic of order 6.) (C) (5) Find the orders of all the subfields of K. VII. (A) (15) State the Fundamental Theorem of Galois Theory. (B) (20) Suppose we know that K is a Galois extension of F with Gal_{F} K an abelian group of

order 18 whose Sylow 2- and 3-subgroups are both cyclic. How many intermediate fields E, F ⊂ E ⊂ K, are there and what are each of their degrees over F?

VIII. Let p be a prime number and let

ζ_{p} = cos (\frac{2π}{p} ) + i sin (\frac{2π}{p} ) ,

be the primitive pth root of 1 in ℂ. (A) (10) Show that if F is a field containing ζ_{p} and K is the splitting field of the polynomial

x^{p} − a ∈ F [x] for some a ∈ F that is not the pth power of any element of F, then Gal_{F} K is isomorphic to the additive group (ℤ_{p}, +) (cyclic of order p).

(B) (10) Show that Gal_{ℚ}ℚ(ζ_{p}) is isomorphic to the multiplicative group (ℤ_{p}^{∗}, ·) (cyclic of order p − 1).

IX. Let ζ_{p} be as in question VIII, and let K be the splitting field of x^{p} − 2 ∈ ℚ[x]. (A) (10) Show carefully that [K : ℚ] = p(p − 1). (B) (5) Is Gal_{ℚ}K a solvable group? Why or why not?

Have a safe, productive, and enjoyable summer!