Problem Set #4 in Math 114: Fields, Vector Spaces, and Fibonacci Sequences, Assignments of Abstract Algebra

Download Problem Set #4 in Math 114: Fields, Vector Spaces, and Fibonacci Sequences and more Assignments Abstract Algebra in PDF only on Docsity! PROBLEM SET # 4 MATH 114 Due February 16. Read chapter 1 in Artin’s book. We assume that all fields are commutative in homework problems. 1. Let F be a field, and F [i] denote the set of all expressions a + bi, with a, b ∈ F . Define addition and multiplication in F [i] by (a + bi) + (c + di) = (a + c) + (b + d) i, (a + bi) (c + di) = ac− bd + (ad + bc) i. Determine if F [i] is a field for F = Q, R, Z3, Z5. 2. Assume that charF = p. Prove that (a + b)p = ap + bp. Hint: use binomial formula. 3. Prove the little Fermat’s theorem ap ≡ a mod p for any prime p and integer a. Hint: use the previous problem. 4. Let V be a vector space of dimension n and A : V → V be a linear map such that AN = 0 for some integer N > 0. Prove that An = 0. Hint: check that ImAk is a proper subspace in Im Ak−1. 5. Find a formula for a general term of the Fibonacci sequence 1,1,2,3,5,8,13,. . . Hint: write the Fibonacci sequence as a linear combination of 1, α, α2, α3, . . . and 1, β, β2, β3, . . . , where α = 1 + √ 5 2 , β = 1 − √ 5 2 . 6. Let F = Zp. (a) Prove that the number of one dimensional subspaces in F n equals p n −1 p−1 ; (b) (Extra credit) Find the number of 2-dimensional subspaces in F n. Date: February 9, 2006. 1