1. Using Euclid’s algorithm for polynomials, find the gcd of ...

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1. Using Euclid’s algorithm for polynomials, find the gcd of the polynomials p(x) = x 9 − 2x 8 + x 7 + x 5 − 4x 4 + 5x 3 + 15x 2 − 34x + 17 q(x) = x 4 − 2x 2 + 1 2. (a) Consider Z[i] = {a + ib|a, b ∈ Z}, a subring of C. By considering the numbers of C as points in the plane and looking at the shape made by the points which lie in Z[i], prove the following : for every complex number z, we can find a q ∈ Z[i] such that |z − q| ≤ 1, where |a + ib| = √ a 2 + b 2 . (b) For a, b ∈ Z[i], apply part (a) of this exercise to z = a/b to conclude that Z[i] is an euclidean domain. 3. (a) Let R be a domain, let f be a size function on R and let u ∈ R. Prove that u is a unit if and only if f(u) = 1. (Hint : Find f(1) first) (b) Find all units of Z[i] and of F[x], where F is a field. 4. Prove that 4 + i is an irreducible element of Z[i]. (Hint : Use the size function on Z[i] that we discussed in the class) 5. (a) Prove that x 2+1 is an irreducible element of Q[x]. Is it an irreducible element of C[x]? (b) Find a factorization of x 4 + x 2 + 1 into irreducible elements in Q[x]. (Hint : x 4 + x 2 + 1 = (x 4 + 2x 2 + 1) − x 2 ) 6. (a) Let F be a field. Let a, b ∈ F such that a 6= 0. Then prove that ax + b is an irreducible element of F[x]. (b) Let F be a field and R = F[x] is the ring of polynomials with coefficients in F. Suppose that p(x) ∈ F[x] is a polynomial. Let a ∈ F. Then we can define polynomial l(x) = x − a. Show that l|p if and only if p(a) = 0. (c) Use part(b) to prove that x 3 − 2 is irreducible in Q[x]. 7. Suppose that R is an integral domain and that the number of elements of R is finite. Let a ∈ R be a fixed non-zero element. Show that the multiplication-by-a map, g : R → R, g(x) = ax must be injective. Using the fact that the cardinality of R is finite and g is injective, prove that g is surjective. Using this, show that R is a field.

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