Mathematics 351 – Abstract Algebra I Final Examination – December 12, 2007

Directions

Do all work in the blue exam booklet. Do not place anything you wish to have considered for credit on the exam sheet. There are 200 points, distributed as indicated next to the questions.

Notation: In the following, unless specified otherwise, F denotes a field.

I. Terminology

A) (5) What does it mean to say that R is a ring with identity? B) (5) What does it mean to say that a ∈ F is a root of f(x) ∈ F [x]? C) (5) What does it mean to say that f(x) ∈ F [x] is irreducible? D) (5) What does it mean to say that a subset I of a ring R is an ideal in R? E) (5) What does it mean to say that a permutation σ ∈ S_{n} is even?

II.

A) (15) State and prove the Eisenstein Irreducibility Criterion in Q[x]. B) (10) Use the Eisenstein criterion to show that there are irreducible polynomials of

every degree ≥ 1 in Q[x].

III.

A) (10) Show that a ∈ F is a root of f(x) ∈ F [x] if and only if (x − a)|f(x) in F [x]. B) (10) Express f(x) = x^{4} − 16 as a product of irreducible polynomials in Q[x], in R[x],

and finally in C[x]. C) (5) In what sense are your factorizations from part B unique?

IV. Let p(x) = x^{2} + x + 1 ∈ Z_{2}[x].

A) (15) Determine the addition and multiplication tables for the quotient ring R =

Z_{2}[x]/(p(x)). B) (5) Is R a field? Why or why not? C) (5) Show that x + 1 ∈ Z_{2}[x]/(p(x)) is a root of the polynomial y^{2} + y + 1 ∈ Z_{2}[y].

V.

A) (10) State the First Isomorphism Theorem for rings. B) (10) Let φ be the mapping

\begin{matrix}φ : Z[x] & → Z_{3} \\ f(x) & ↦ f(0) mod 3. \end{matrix}

That is, φ takes each polynomial

f(x) = a_{n}x^{n} + ⋯ + a_{1}x + a_{0}

to the class of a_{0} mod 3). Show that φ is a ring homomorphism and show that its kernel is the ideal (x, 3) ⊂ Z[x].

C) (5) Is the ideal (x, 3) in Z[x] maximal? Why or why not?

VI. Let R be a ring with identity.

A) (7.5) Show that the set of units in R is a group under the multiplication operation

from R. B) (10) Determine the multiplication table for the group of units in the ring Z_{8} × Z_{4} =

{(r, s) : r ∈ Z_{8} and s ∈ Z_{4}} (componentwise sum and product). C) (7.5) Is your group from part B cyclic? Why or why not?

VII. Let G be a group, and let H be a subgroup of G. Define

N_{H} = {x ∈ G : xHx^{−1} ⊆ H},

called the normalizer of H in G.

A) (10) Let G = GL(2, R) be the group of 2 × 2 invertible matrices under matrix multiplication and let

H = \{( \begin{matrix}1 & r \\ 0 & 1 \end{matrix} ) : r ∈ R\} .

Show that H is a subgroup of G and determine N_{H} in this case. B) (10) Show that for any subgroup of a general group, H ⊆ G, N_{H} is a subgroup of G

containing H, and that H is a normal subgroup of N_{H}. C) (5) Show that the quotient group N_{H}/H for G, H as in part A is isomorphic to the

group R^{∗} × R^{∗}, where R^{∗} denotes the multiplicative group of nonzero real numbers.

VIII.

A) (5) By Lagrange’s theorem, what are the possible orders of subgroups of S_{4}? B) (10) Show that the subset G ⊂ S_{4} defined by

G = {(), (1234), (13)(24), (1432), (24), (1234)(24), (13)(24)(24), (1432)(24)}

is a subgroup of S_{4}. Note that the second four elements are equal to the compositions of the first four with the two-cycle (24).

C) (10) Determine the left and right cosets of G in S_{4}. Is G a normal subgroup of S_{4}? Why or why not?

Have a peaceful and joyous holiday season!