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9. Let F be a finite field with q elements, and let m | q - 1. (a) Prove that F* has a unique subgroup of order m. (b) Let a € F*. Prove that the following are equivalent: (i) a is an mth power in F; i.e., a = ßm for some 3 € F*. (ii) a (9-1)/m = 1. This is known as Euler's criterion. (c) Suppose that q is odd. Prove that -1 is a square in F* q = 1 (mod 4).

9. Let F be a finite field with q elements, and let m | q - 1. (a) Prove that F* has a unique subgroup of order m. (b) Let a € F*. Prove that the following are equivalent: (i) a is an mth power in F; i.e., a = ßm for some 3 € F*. (ii) a (9-1)/m = 1. This is known as Euler's criterion. (c) Suppose that q is odd. Prove that -1 is a square in F* q = 1 (mod 4).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.4: Zeros Of A Polynomial
Problem 33E: Let where is a field and let . Prove that if is irreducible over , then is irreducible over .
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8.9. Let F be a finite field with q elements, and let m | q - 1. (a) Prove that F* has a unique subgroup of order m. (b) Let a € F*. Prove that the following are equivalent: (i) a is an mth power in F; i.e., a = 3m for some ß F*. (ii) a (9-1)/m = 1. This is known as Euler's criterion. (c) Suppose that q is odd. Prove that -1 is a square in F* (Hint. Use (b) with m = 2.) q = 1 (mod 4).
Transcribed Image Text:8.9. Let F be a finite field with q elements, and let m | q - 1. (a) Prove that F* has a unique subgroup of order m. (b) Let a € F*. Prove that the following are equivalent: (i) a is an mth power in F; i.e., a = 3m for some ß F*. (ii) a (9-1)/m = 1. This is known as Euler's criterion. (c) Suppose that q is odd. Prove that -1 is a square in F* (Hint. Use (b) with m = 2.) q = 1 (mod 4).
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