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
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ISBN:9780470458365
Author:Erwin Kreyszig
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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).
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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