Q2for all ?,?,?,?,?∈ℤ:(?^2+??^2)(?^2+??^2)=(??±???)^2+?(??∓??)^2. Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N cana² + nb² (be written as q =z2 + nw? (w, x, y, z E Z), then the integerNcan also be written asNa, b e Z).

We will use standard division algorithms to deduce the given statement.

Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N can be written as q = z² + nw² (w, x, y, z E Z), then the integer can also be written as a? + nb² ( а, bE Z).

Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N can be written as q = z² + nw² (w, x, y, z E Z), then the integer can also be written as a? + nb² ( а, bE Z).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 20E: 20. If and are nonzero integers and is the least common multiple of and prove that.

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Question

for all ?,?,?,?,?∈ℤ:

(?^2+??^2)(?^2+??^2)=(??±???)^2+?(??∓??)^2.

Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N can
a² + nb² (
be written as q =
z2 + nw? (w, x, y, z E Z), then the integer
N
can also be written as
N
a, b e Z).

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