UNIVERSITY OF DUBLIN 1417

TRINITY COLLEGE

Faculty of Science

school of mathematics

JS Mathematics SS Mathematics Trinity Term 1994

Course 373

Exam Hall

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 A

PLEASE INSERT Michael Purser’s questions here.

SECTION B

7. Show that the multiplicative group F^{×} = F − {0} of a finite field F is cyclic. Determine all the primitive elements (multiplicative generators) in GF(17).

8. Listing the elements of GF(8) 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.

Show further 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.

Prove that each subfield K ⊂ F is of the form

K = {x ∈ F : Φ^{m}(x) = x}

for some m.

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10. Show that if f(x) is an irreducible polynomial of degree d over GF(p^{d}), then

f(x) | x^{p^{d}} − x.

Hence or otherwise show that if there are N(d, p) prime polynomials of degree d over GF(p^{n}) then

∑_{d|n} dN(d, p) = p^{n}.

Determine the number of prime polynomials of degree 6 over GF(2), and find one of them.

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