non quadric residue of Q/Z

bhelmes · 2022-06-28, 20:48

A peaceful and joyful night for you

if you consider the finite group of rationals modulo p resp. u/v mod p element of Q/Z

Is u/v a non quadratic residue,
if jacobi (u²+v², p)=-1 or
if jacobi (u, p)*jacobi (v, p)=-1

A short answer, perhaps with a little explication, would be very nice and perhaps also interesting for other members.

MattcAnderson · 2022-07-01, 06:51

Hi bhelmes and others,
You must know about finite cyclic groups.
For example, the simplest group contains only 1 element.
We can call this the identity element, or shorten it to
just one letter 'e'.

The second easiest group is like the odd and the even numbers.
Let's use 'o' for odd and 'e' for even.
We have a finite group under addition.

Other (mathematical) groups can be formed under multiplication,
or a specialty operation

Cheers

https://reference.wolfram.com/language/guide/GroupTheory.html

https://www.youtube.com/watch?v=zkADn-9wEgc

You may need a reference on Abstract Algebra
I no longer have all of my textbooks.
I'm 'living the life of Riley'.

2022-06-28, 20:48	#1
bhelmes Mar 2016 2²·101 Posts	non quadric residue of Q/Z A peaceful and joyful night for you if you consider the finite group of rationals modulo p resp. u/v mod p element of Q/Z Is u/v a non quadratic residue, if jacobi (u²+v², p)=-1 or if jacobi (u, p)jacobi (v, p)=-1 A short answer, perhaps with a little explication, would be very nice and perhaps also interesting for other members. Last fiddled with by bhelmes on 2022-06-28 at 20:49*

2022-07-01, 06:51	#2
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 5×233 Posts	introduction to mathematical groups Hi bhelmes and others, You must know about finite cyclic groups. For example, the simplest group contains only 1 element. We can call this the identity element, or shorten it to just one letter 'e'. The second easiest group is like the odd and the even numbers. Let's use 'o' for odd and 'e' for even. We have a finite group under addition. Other (mathematical) groups can be formed under multiplication, or a specialty operation Cheers https://reference.wolfram.com/language/guide/GroupTheory.html https://www.youtube.com/watch?v=zkADn-9wEgc You may need a reference on Abstract Algebra I no longer have all of my textbooks. I'm 'living the life of Riley'.

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