Mathematics 351 – Abstract Algebra I Midterm Exam 1 – October 1, 2007

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 possible points.

I. Terminology. A) (5) Give an example of a ring R that is not a commutative ring. B) (5) What does it mean to say that R is a ring with identity? C) (5) What does it mean to say that u ∈ R is a unit? D) (5) Is every commutative ring with identity an integral domain? If so, say why. If not,

give a counterexample.

II. Let R be a ring. Let b be any one particular element of R, and let

C_{b} = {a ∈ R : ab = ba}.

A) (10) Show that C_{b} is a subring of R. B) (10) Let S be a second ring, and let φ : R → S be a ring homomorphism. Show that

if b ∈ R, then φ(C_{b}) ⊆ C_{φ(b)} ⊆ S.

III. Let F [x] be the ring of polynomials with coefficients in a field F. A) (10) Show that the quotient and remainder under the division of f(x) by g(x) in F [x]

are unique. B) (5) Define: f(x) is irreducible in F [x]. C) (10) In what sense are factorizations of polynomials into irreducibles unique? (State

the theorem that answers this question; you do not need to prove it.)

IV. A) (10) Let F be a field. 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) Let F be a field. If a_{1}, …, a_{n} are distinct roots of f(x) ∈ F [x] in F, show that

((x − a_{1})(x − a_{2}) ⋯ (x − a_{n})) |f(x). (Hint: induction on n). C) (5) What are the roots of x^{2}+2x ∈ Z_{8}[x] in Z_{8}? Why is your result not a contradiction

to part B?

V. All parts of this question refer to

f(x) = 5x^{5} − 12x^{3} + 36x^{2} + 18 ∈ Q[x].

A) (5) List all rational numbers that could be roots of f according to the Rational Root

Test. B) (5) Use any applicable method to decide whether f is irreducible in Q[x], and state

which criterion or criteria you are using.