1. For cach of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) If F is a field and x, y E F are nonzero, then x | y in F. (b) Let d e Z and suppose that vd ¢ Q. If 71 and n2 are primes in Z[Vd] such that T1 T2, then T1 = T2. (c) Let m e N. If a E Z satisfies a? = 1 (mod m), then a = 1 (mod m) or a = -1 (mod m).
1. For cach of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) If F is a field and x, y E F are nonzero, then x | y in F. (b) Let d e Z and suppose that vd ¢ Q. If 71 and n2 are primes in Z[Vd] such that T1 T2, then T1 = T2. (c) Let m e N. If a E Z satisfies a? = 1 (mod m), then a = 1 (mod m) or a = -1 (mod m).
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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![1.
For cach of the following statements, indicate whether the statement is true
or false and justify your answer with a proof or a counterexample.
(a) If F is a field and x, y E F are nonzero, then x | y in F.
(b) Let d e Z and suppose that vd ¢ Q. If 71 and n2 are primes in Z[Vd] such that
T1| T2, then T1 = T2.
(c) Let m e N. If a e Z satisfies a? = 1 (mod m), then a = 1 (mod m) or a = -1
(mod m).
(d) If a is an odd integer, then either a = 1 (mod 4) or a = 3 (mod 4).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe664530a-7a3a-4495-bab4-46da3ec55b5f%2F5309441e-29a1-46da-a5cb-ff23a8aade05%2F1cgkpeg_processed.png&w=3840&q=75)
Transcribed Image Text:1.
For cach of the following statements, indicate whether the statement is true
or false and justify your answer with a proof or a counterexample.
(a) If F is a field and x, y E F are nonzero, then x | y in F.
(b) Let d e Z and suppose that vd ¢ Q. If 71 and n2 are primes in Z[Vd] such that
T1| T2, then T1 = T2.
(c) Let m e N. If a e Z satisfies a? = 1 (mod m), then a = 1 (mod m) or a = -1
(mod m).
(d) If a is an odd integer, then either a = 1 (mod 4) or a = 3 (mod 4).
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