10 a. Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i). b. Show that an odd prime p in Z is irreducible in Zliif and only if p = 3 (mod 4). (Use Theorem 47.10.) 11 Prove Lemma 47.2. thot N of Fxamnle 47 9 is multinlicative, that is, that N(aB) = N (a)N(B) for a, BEZIV-5].
10 a. Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i). b. Show that an odd prime p in Z is irreducible in Zliif and only if p = 3 (mod 4). (Use Theorem 47.10.) 11 Prove Lemma 47.2. thot N of Fxamnle 47 9 is multinlicative, that is, that N(aB) = N (a)N(B) for a, BEZIV-5].
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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Section 47 number 10 (a)and (b). Use theorem 47.10
Transcribed Image Text:to a sum of
sauar
o have now AY auesion for the
only even prime
(Fermat's p = a² + b² Theorem) Let p be an odd prime in Z. Then p = a² + b² for
integers a and b in Z if and only if p = 1 (mod 4).
47.10 Theorem
![of D.
among all |N(B)| > 1 for BE D.Show that n is an irreducible
10 a. Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i].
b. Show that an odd prime p in Z is irreducible in Zliif and only if p = 3 (mod 4). (Use Theorem 47.10.)
1 Prove Lemma 47.2.
Duun thot N of Fxamnle 47 9 is multiplicative, that is, that N(aß) = N(a)N(B) for a, BE Z[v-5).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4dd8e23-ab66-4b24-8e54-a64daec9031c%2Fc4ad8c64-b201-4772-b86f-37fc2c5b58d2%2Fx42lmq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:of D.
among all |N(B)| > 1 for BE D.Show that n is an irreducible
10 a. Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i].
b. Show that an odd prime p in Z is irreducible in Zliif and only if p = 3 (mod 4). (Use Theorem 47.10.)
1 Prove Lemma 47.2.
Duun thot N of Fxamnle 47 9 is multiplicative, that is, that N(aß) = N(a)N(B) for a, BE Z[v-5).
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