Rings and Fields

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## 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.

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.

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).

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.

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.

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?

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?

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.

SOLUTIONS TO WHOLE SET
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JOC/EFR 2004