**Unformatted text preview: **09/15/2000 FRI 16:33 FAX 6434330 MOFFITT LIBRARY .001 Math 113 FINAL EXAM
May 13, 1996 Prof. Wu 1. (5%) Prove that for an integer n, 3 | n i=r 3 I (sum of digits of n). 2. (5%) Let ﬂu) = 2:" + a..-1z"“1 + - - 411:: + an be a polynomial with integer coefﬁcients, and let :- be
a rational number such that f(r) = 0. Show that 1' has to be an integer and r I no. 3. (5%) Find a. minimal polynomial of m over 0. (Be sure to prove that it is minimal.) 4. (5%) Let n be a positve integer __>_ 2 such that u | (6"“1 — 1) for all integers b which are not a multiple of 11. What can you say about u? 5. (5%) Do the nonzero elements of 213 form a cyclic group under multiplication? Give reasons.
6. (10%) Let p be a prime.
(i) for k :— 1.---p-— 1, where (in?) E Kalli—Bl-
(b) Prove: the mapping f : Zp «4 Z, deﬁned by f(k) = 1:? for all I: E Z, is a ﬁeld isomorphism.
7. (10%) Is 2:4 + 22: + 3 irreducible over 1R? Is it irreducible over Q? Give reasons.
8. (10%) Let F E {a+ ib : a, b E Q} and let K E (Dbl/(1:2 + 1)Q{x]. Show that F is isomorphic to K
as ﬁelds by deﬁning a map (,0 : F -—> K and show that 4; has all the requisite properties.
9. (10%) If I? is a root of 2:3 —- 2+ 1, ﬁnd some 17(2) 6 QM so that (,5!2 —- 2)p(ﬁ) .—'. 1.
10. (10%) Let C = elk/3. Compute (Q(C, 3/5) :Q(§)).
11. (125%) (In (sf-(d) below, each part could be done independently.)
(3) Assume that if p is a prime, then 4:94 + zap—2 + - - - + 1 is irreducible over Q. Compute
(Q(coﬁ2dr/T) + i sin(27r/ 7)) : Q). (Each step should be clearly explained.)
(13) Suppose the regular 7-gon can be constructed with straightedge and compass. Explain why
(Q(cos(21r/7) : Q) = 2" for some I: 6 2*.
(c) If F E Q(cos(27r/7)), show that (F(isin(21r/7)) : F) = 1 or 2.
(d) Use (b) and (c) to conclude that if the regular 7-gon can be constructed with straightedge and compass,
then (Q(cos(27r/7) + isin(21r/7)) : Q) = 2"" for some m E 2". (e) What can you conclude from (a) and (d)? What is your guess concerning the construction of the regular (a) Prove: p ll-gon, the regular Iii-gen, the regular 23-gon, etc? ...

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