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Unformatted text preview: Mathematics 115 Final Exam, May 18, 1994 Professor K. A. Ribet 1. Let A be a ﬁnite abelian group. If ϕ : A → C∗ is a non-trivial homomorphism, show that ϕ(a) = 0.
a∈A 2. Calculate the number of solutions to the congruence x3 ≡ 8 mod 5040. (Note that 5040 = 7!.) 3. A pseudoprime is a composite integer p for which 2p ≡ 2 mod p. Show that 11 · 31 is a n pseudoprime. Prove that every Fermat number 22 + 1 is either a prime or a pseudoprime. [The ﬁrst known even pseudoprime 161038 was found by Berkeley’s D. H. Lehmer in 1950.] 4. Suppose that p is an odd prime and that a is an integer prime to p. Gauss’s lemma a = (−1)µ . Explain the deﬁnition of the quantity µ may be summarized as the identity p 2 that appears in this formula. Use Gauss’s lemma to compute when p ≡ 1 mod 8. p
∞ 1 for s > 1. Prove that ζ (s) − 1 < t−s dt < ζ (s) by ns 1 n=1 using techniques from ﬁrst-year calculus. Use these inequalities to prove that the limit lim+ (s − 1)ζ (s) exists and equals 1. (In this problem, s is always a real number.) ∞ 5. Recall that ζ (s) = s→1 6. Let D = Z[ω ], where ω is a complex third root of 1. a. Show that the unit group of D has order 6. b. If u is a unit of D which is congruent to 1 mod (3), prove that u = 1. 7. Let ζ = e2πi/p , where p ≥ 3 is prime. Show that a mod p 8. Let p be an odd prime, and let J be the Jacobi sum
a∈Fp aa ζ= p ζa . a mod p 2 ψ (a)ϕ(1 − a), where ψ and ϕ are non-trivial characters F∗ → C∗ . p a. What value did we ﬁnd for J in case ψϕ is the trivial character? b. What expression did we obtain for J if ψϕ is not the trivial character? c. Show that every prime congruent to 1 modulo 4 is the sum of two integral squares. This is a closed book exam. Time limit: three hours. ...
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This note was uploaded on 10/31/2009 for the course STAT 131A taught by Professor Isber during the Spring '08 term at University of California, Berkeley.
- Spring '08