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

  1. Home
  2. Homework Library
  3. Mathematics
  4. Algebra
  5. 1. Using Euclid’s algorithm for polynomials, find the gcd of ...


Transcribed TextTranscribed Text

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.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Algebra Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats