Mathematics 352 – Abstract Algebra II Midterm Exam 2 – April 16, 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) What does it mean to say that a polynomial f(x) ∈ F [x] is separable?

(B) (5) What is an F -automorphism of a field K containing a subfield F?

(C) (5) What does it mean to say that an extension field K of a field F is Galois over F?

II.

(A) (5) State a necessary condition for the real number r to be constructible by straightedge and compass, starting from two given points (0, 0) and (1, 0).

B) (10) Given a sphere of radius 1, is it possible to construct the radius of a sphere with 5 times the volume of the original sphere? Why or why not?

III. Let K be a finite field of order 81.

(A) (5) What is the characteristic of K?

(B) (10) Show that the mapping φ : K → K given by φ(x) = x^{3} is an automorphism of K.

(C) (5) Does K have a subfield of order 27? Why or why not?

IV.

(A) (15) State the Fundamental Theorem of Galois Theory.

(B) (10) Show that if K is a finite dimensional extension of F, H is a subgroup of Gal_{F} K, and E is the fixed field E = E_{H}, then K is a simple, normal, separable extension of E.

V. Let K be a Galois extension of F with Gal_{F} K = ⟨σ⟩ a cyclic group of order 12.

(A) (10) What would the Galois correspondence look like in this case? (Construct the parallel diagrams of subgroups and intermediate fields, showing inclusions, giving the intermediate fields other than F and K “generic” names like E_{1}, E_{2}, etc.)

(B) (5) What are the degrees of each of the intermediate fields over F?

(C) (10) Let E = E_{H} for H = ⟨σ^{3}⟩. Is E a normal extension of F? Why or why not? What is Gal_{F} E?

Extra Credit (10) Let K be the splitting field of the polynomial f(x) = x^{3} −2 ∈ ℚ[x]. By Discussion 2, we know that G = Gal_{ℚ}K ≃ S_{3}. What is the fixed field E_{H} if H is the cyclic subgroup of G of order 3? (In S_{3}, this is the subgroup generated by the permutation (123).) Compute the fixed field using a basis for K over ℚ.