UNIVERSITY OF DUBLIN 1417

TRINITY COLLEGE

Faculty of Science

school of mathematics

JS Mathematics SS Mathematics Trinity Term 1996

Course 373

Saturday, June 8  G.M.B.  09.30 — 12.30

Dr. M. Purser and Dr. T.G. Murphy

Answer Section A and B in separate answer books. Section A - Answer 3 out of the 6 questions Section B - Answer 2 out of the 4 questions

SECTION B

7. Prove that the multiplicative group F^{×} = F − {0} of a finite field F is cyclic.

Find all the generators of GF(17)^{×}. How many generators does GF(64)^{×} possess?

8. Listing the elements of GF(9) in any way you wish, draw up the addition and multiplication tables for this field.

9. Define the characteristic of a field, and show that the characteristic of a finite field F is always a prime number.

Show that a finite field F of characteristic p contains p^{n} elements for some n.

Let F be a finite field of characteristic p. Show that F contains p^{n} elements for some n. Show also that the map

Φ : x ↦ x^{p}

is an automorphism of F; and show that every automorphism of F is of the form x ↦ Φ^{i}(x) for some i.

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10. Show that the prime (irreducible) factors of the polynomial

x^{p^{n}} − x

over GF(p) are precisely the prime polynomials over GF(p) whose degree d divides n. Hence show that if there are Π(n) = Π_{p}(n) prime polynomials of degree n over GF(p), then

∑_{d|n} dΠ(d) = p^{n}

for each positive integer n. Determine all prime polynomials of degree 5 over GF(2). Explain what is meant by saying that a prime polynomial over GF(p) is primitive. Show that there are

\frac{1}{n}ϕ (p^{n} − 1)

primitive polynomials of degree n over GF(p), where ϕ is Euler’s function. Determine which of the prime polynomials of degree 5 over GF(2) are primitive.

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