Use the Euclidean Algorithm to compute the greatest common divisors of the following

Use the Euclidean Algorithm to compute the greatest common divisors of the following pairs of polynomials over Q. Also express each greatest common divisor as a linear combination of the two given polynomials (as in Theorem 36.2). x 3 - 3x2 + 3x - 2 and x 2 - 5x + 6

Use the Euclidean Algorithm to compute the greatest common divisors of the following pairs of polynomials over Q. Also express each greatest common divisor as a linear combination of the two given polynomials (as in Theorem 36.2). X4 + 3x 2 + 2 and x 5 - x

Use the Euclidean Algorithm to compute the greatest common divisors of the following pairs of polynomials over Q. Also express each greatest common divisor as a linear combination of the two given polynomials (as in Theorem 36.2). x 3 + x 2 - 2x - 2 and X4 - 2x 3 + 3x2 - 6x

Use the Euclidean Algorithm to compute the greatest common divisors of the following pairs of polynomials over Q. Also express each greatest common divisor as a linear combination of the two given polynomials (as in Theorem 36.2). x 5 + 4x and x 3 - x.

Determine the monic associate of 2x3 - x + 1 E Q[x 1.

Determine the monic associate of 1 + x - ix2 E iC[x1.

Determine the monic associate of 2x5 - 3x2 + 1 E Z7[x1.

Determine the monic associate of2x5 - 3x2 + 1 E Z5[x1.

Verify that x3 - 3 E Z7[X] is irreducible.

Verify that is reducible.

Write x 3 + 3x2 + 3x + 4 E Z5 [x] as a product of irreducible polynomials.

Write x 5 + X4 + x 2 + 2x E Zl [x] as a product of irreducible polynomials.

Prove that the polynomial lAx) in the proof of Theorem 36.1 satisfies both requirements (a) and (b) of the theorem

Prove the uniqueness of d(x) in Theorem 36.1

Prove Theorem 36.2. (The remark preceding it suggests the method.)

Explain why each nonzero polynomial has preCisely one monic polynomial among its associates.

Write the proof of the second corollary of Theorem 36.2.

Complete the proof of the Unique Factorization Theorem

(a) Prove that (x - I) I I(x) in Z2[X] iff I(x) has an even number of nonzero coefficients. (b) Prove that if deg I(x) > 1 and I(x) is irreducible over Z2, then I(x) has constant term I and an odd number of nonzero coefficients. (c) Determine all irreducible polynomials of degree 4 or less over Z2. (d) Write each polynomial of degree 3 over Z2 as a product of irreducible factors.

(a) By counting the number of distinct possibilities for (x - alex - b), verify that there are pep + 1)/2 monic reducible polynomials of degree 2 over Zp (p a prime). (b) How many monic irreducible polynomials of degree 2 over Zp are there?

State and prove a theorem establishing the existence of a unique least common multiple for every pair of polynomials, not both the zero polynomial, over a field F. (Compare Theorem 12.3 and Theorem 36.1.)

(Eisenstein's irreducibility criterion) Assume that p is a prime, I(x) = ao + alX + ... + a"x" E Z[x], pia; (0 =:: i =:: n - 1), p21' ao and p l' a". Then I(x) is irreducible in Z[x]. Give an indirect proof of this by justifying each of the following statements. (a) Assume that I(x) = (bo + b1x + ... + b"x")(co + CIX + ... + c,x'). Then p does not divide both bo and co. (b) But p divides one of bo and co. Assume that p l' bo and p !co. (c) Since p1' c, (why?), not all Cj are divisible by p. Let k be the smallest integer such that P1'Ck andplcj forO =:: j <k. (d) Because ak = bkCo + bk_lCI + .,. + bock. we can conclude that p I bOCk> which is a contradiction. (For applications, see Problems 36.23 and 36.24.)

Use Eisenstein's irreducibility criterion (Problem 36.22) to show that each of the following polynomials is irreducible in Z[x]. (a) xl + 6x2 + 3x + 3 (b) x 5 - 5x3 + 15

Use Eisenstein's irreducibility criterion (Problem 36.22) to prove that if p is a prime, then the polynomial x p - 1 f(x) = -- = x p- I +xp - 2 + ... +x + 1 x - I is irreducible in Z[xl. (Suggestion: Replace x by y + 1, and let g(y) denote the result. Show that g(y) is irreducible in Z[yl, and explain why this implies that f(x) is irreducible in Z[xl. The polynomial f (x) is called a cyclotomic polynomial; if p is odd, its roots are the imaginary pth roots of unity.)