UNIVERSITY OF DUBLIN 1417

TRINITY COLLEGE

Faculty of Science

school of mathematics

JS Mathematics SS Mathematics Hillary Term 2000

Course 373 Part A

Friday, March 10  Maxwell Theatre  16.15 — 17.45

Dr. T.G. Murphy

Attempt 5 questions. (If you attempt more, only the best 5 will be counted.) All questions carry the same number of marks.

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

Find all the generators of (ℤ/13)^{×}.

2. 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 also that any subfield of F contains p^{m} elements, where m | n

3. Listing the elements of 𝔽_{8} in any way you wish, draw up the addition and multiplication tables for this field.

4. Prove that two finite fields containing the same number of elements are necessarily isomorphic.

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5. If F is a finite field of characteristic p show 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.

6. Show that if there are Π(n) = Π_{p}(n) prime polynomials of degree n over 𝔽_{p}, then

∑_{d|n} dΠ(d) = p^{n}

for each positive integer n. Determine all prime polynomials of degree 5 over 𝔽_{2}.

7. Prove Wedderburn’s Theorem, that every finite skew field is in fact commutative.

\begin{matrix}c \\ ◯ \end{matrix} UNIVERSITY OF DUBLIN 2002