Name: Student number: METU MATH 116, Final Examination _{1 2 3 4 5 6 7 Σ}

Date & time. June 7, 2011, 13:30–15:30 (120 minutes). Instructors. Finashin, Pamuk, Pierce, Solak.

Problem 1 (12 points). Is x^{3} + x + 1 reducible over:

(a) ℚ ? Solution. f(x) = x^{3} + x + 1 has degree 3 ⟹ f is reducible if and only if f has a root (it concerns parts (a),(b), and (c)).

\frac{p}{q} ∈ ℚ is a root ⟹ p|1, q|1 ⟹ \frac{p}{q} = ±1.

f(1) = 3, f(−1) = −1 ⟹ f(x) has no roots ⟹ f is irreducible

(b) ℤ_{3} ? Solution. f(1) = 0 ∈ ℤ_{3}, ⟹ f has a root ⟹ f is reducible (divisible by (x − 1)).

(c) ℤ_{5} ? Solution. f(0) = 1, f(1) = 3, f(2) = 11, f(3) = 31, f(4) = 69.

No zeros modulo 5 ⟹ f is irreducible over ℤ_{5}.

Problem 2 (8 points). Letting f(x) = 3x^{4} + 5x^{3} + x^{2} + 5x − 2, write f(x) as a product of irreducible polynomials over ℚ.

Solution. \frac{p}{q} is a root of f ⟹ p|2, q|3 ⟹ p ∈ \{±1, ±2\}, q ∈ \{±1, ±3\} ⟹ \frac{p}{q} ∈ \{±1, ±2, ±\frac{1}{3}, ±\frac{2}{3}\}.

f(1) = 3 + 5 + 1 + 5 − 2 ≠ 0, f(−1) = 3 − 5 + 1 − 5 − 2 ≠ 0, f(2) = 48 + 40 + 4 + 10 − 2 ≠ 0, f(−2) = 48 − 40 + 4 − 10 − 2 = 0.

−2 is a root ⟹ f(x) is divisible by (x + 2). Using division algorithm, f(x) = (x + 2)(3x^{3} − x^{2} + 3x − 1) \frac{1}{3} is a root of 3x^{3} − x^{2} + 3x − 1

3x^{3} − x^{2} + 3x − 1 = (3x − 1)(x^{2} + 1).

\underline{Answer:} f(x) = (x + 2)(3x − 1)(x^{2} + 1) (x + 2), (3x − 1) have degree 1 ⟹ irreducible x^{2} + 1 > 0 for x ∈ ℚ ⟹ no rational roots ⟹ irreducible.

Problem 3 (5 points). Is every integral domain a field? Explain.

Problem 4 (15 points). Find a polynomial f(x) of least positive degree with the given properties. (Your answer should show the coefficients of f(x).)

(a) f(x) is over ℂ, and f(2i) = 0 = f(1 + i).

(b) f(x) is over ℝ, and 2i and 1 + i are zeros of it.

(c) f(x) is over ℤ_{2}, and 1 (that is, [1]) is a zero of multiplicity 4. Answers.

(a) f(x) =

(b) f(x) =

(c) f(x) = Problem 5 (7 points). Over a field K, suppose f(x) is a polynomial with no zeros in K. Must f(x) be irreducible over K? Explain.

Problem 6 (13 points). Working over ℤ_{3}, letting

f(x) = x^{5} + x + 1,  g(x) = x^{2} + 1,

find s(x) and t(x) such that f(x) · s(x) + g(x) · t(x) = 1.

Problem 7 (20 points). Let R be the subring {x + yi: x, y ∈ ℤ} of ℂ, and let I be the ideal {x + yi: x, y ∈ 2ℤ} of R.

(a) How many additive cosets has I in R? List them clearly.

(b) Is the quotient R/I cyclic as an additive group? Explain.

(c) Show that the function ϕ from R to ℤ_{2} given by

ϕ(x + yi) = [x + y]  (∗)

is a ring homomorphism.

(d) Does the same formula (∗) define a ring homomorphism from R to ℤ_{3}? Explain.