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
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 .
Related questions
Question
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps with 5 images
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning