Rings and Fields

Previous page
(Exercises 4)
Contents Next page

Exercises 5

  1. We have seen that the polynomials x2 + 1, x2 + 2x + 2 ∈ Z3[x] are irreducible. Find a third irreducible monic quadratic polynomial and prove that the field it defines is isomorphic to the field determined by the other two.
    The polynomials x3 + x2 + 1, x3 + x + 1 ∈ Z2[x] are irreducible. Find an isomorphism between the two fields of order 8 they determine.

    Solution to question 1

  2. Prove that if p is prime there are 1/2 p(p - 1) irreducible quadratic monic polynomials in Zp[x].
    Find a formula for the number of irreducible cubic monic polynomials.

    Solution to question 2

  3. Let F be a field of characteristic p. Prove that the map rrp is a ring homomorphism.
    [You will need to use the fact that if p is prime the binomial coefficient pCr is divisible by p for 0 < r < p. Can you prove this?]
    The above is called the Frobenius map introduced by the German mathematician Georg Frobenius (1849 to 1917).

    Solution to question 3

  4. Let f be a ring homomorpism from R onto S with kernel the ideal I. Prove that f defines a one-one correspondence between the ideals of R which contain I and the ideals of S.
    This result is called the Correspondence Theorem for Rings.

    Solution to question 4

  5. Let R be the ring of 2 × 2 matrices with entries in a field F. Verify that . Find other similar expressions and deduce that the two-sided ideal generated by a single matrix is either {0} or the whole ring.
    A ring with only these two ideals is called simple.

    Solution to question 5

  6. Find an element with multiplicative order 8 in the field Z3[x]/ < x2 + 1 > and deduce that the ring of units is indeed cyclic.
    Find the inverse of the element x in the field Z3[x]/ < x3 + 2x + 1 > of order 27.
    Hence or otherwise find the multiplicative order of the element x in this field.
    How many elements with multiplicative order 26 are there in this field?

    Solution to question 6

  7. Let R be the ring Z6[x]. How many roots does the polynomial x2 + x have in R? How can this happen with a quadratic?

    Solution to question 7

  8. Fermat's Little Theorem states that if p is prime then ap = a mod p for any integer a. Prove that the polynomial xp - x has p roots in Zp .
    Prove that the polynomial xpk- x has pk roots in a field with pk elements.

    Solution to question 8

Previous page
(Exercises 4)
Contents Next page

JOC/EFR 2004