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8.8. Let F be a finite field of order q, and assume that q is odd. (a) Let a, b € F*. If a² = b², prove that either a = b or a = -b. (b) Show by way of an example that (a) is not true for the rings Z/8Z and Z/15Z. (c) Let R = {a² : a € F*} and N = {b € F* : b & R} be, respectively, the set of squares and non-squares in F* 21 Prove that R and N each contain exactly (q-1)/2 distinct elements.

8.8. Let F be a finite field of order q, and assume that q is odd. (a) Let a, b € F*. If a² = b², prove that either a = b or a = -b. (b) Show by way of an example that (a) is not true for the rings Z/8Z and Z/15Z. (c) Let R = {a² : a € F*} and N = {b € F* : b & R} be, respectively, the set of squares and non-squares in F* 21 Prove that R and N each contain exactly (q-1)/2 distinct elements.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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8.8. Let F be a finite field of order q, and assume that q is odd. (a) Let a, b € F*. If a² = b², prove that either a = b or a = : -b. (b) Show by way of an example that (a) is not true for the rings Z/8Z and Z/15Z. (c) Let R = {a² : a € F*} and N = {b € F* : b & R} 21 be, respectively, the set of squares and non-squares in F*.2¹ Prove that R and N each contain exactly (q − 1)/2 distinct elements. (d) Let f(x) be the polynomial 9-1 f(x) = = x²=¹²¹ - 1. 2 Prove that R is exactly the set of roots of f(x) in F. (Hint. Use Lagrange to prove that the elements of R are roots. Then use (c) and Theorem 8.8(c).) (e) Let c E F*. Prove that 9-1 C 2 = (1 -1 if c = Q, if c E N. (Hint. Lagrange says that every element of F* is a root of x-1 1. Factor this polynomial as f(x) g(x) and use (d).) (f) Let a1, a2 € R and b₁, b2 EN. Prove that a₁a2 ER and b₁ b₂ € R. The first of these facts is hardly surprising, since indeed, the product of two squares is a square in any commutative ring. But the second fact is surprising, since in most rings, most products of non-squares won't be squares.
Transcribed Image Text:8.8. Let F be a finite field of order q, and assume that q is odd. (a) Let a, b € F*. If a² = b², prove that either a = b or a = : -b. (b) Show by way of an example that (a) is not true for the rings Z/8Z and Z/15Z. (c) Let R = {a² : a € F*} and N = {b € F* : b & R} 21 be, respectively, the set of squares and non-squares in F*.2¹ Prove that R and N each contain exactly (q − 1)/2 distinct elements. (d) Let f(x) be the polynomial 9-1 f(x) = = x²=¹²¹ - 1. 2 Prove that R is exactly the set of roots of f(x) in F. (Hint. Use Lagrange to prove that the elements of R are roots. Then use (c) and Theorem 8.8(c).) (e) Let c E F*. Prove that 9-1 C 2 = (1 -1 if c = Q, if c E N. (Hint. Lagrange says that every element of F* is a root of x-1 1. Factor this polynomial as f(x) g(x) and use (d).) (f) Let a1, a2 € R and b₁, b2 EN. Prove that a₁a2 ER and b₁ b₂ € R. The first of these facts is hardly surprising, since indeed, the product of two squares is a square in any commutative ring. But the second fact is surprising, since in most rings, most products of non-squares won't be squares.
Also note that there's a typo in part (e): What you should prove is that = {'₁₁ (The book has Q where it should have R.) c(q-1)/2 = if c ER, -1 if c E N.
Transcribed Image Text:Also note that there's a typo in part (e): What you should prove is that = {'₁₁ (The book has Q where it should have R.) c(q-1)/2 = if c ER, -1 if c E N.
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