Find Quadratic Numbers in Z3 - Marin's Help Request

[Marin]

Hi there!

I need a little help from your side. Could you give me an example of a quadratic number in Z₃

there is a lemma, that says there are exactly two quadratic numbers there, but I somehow cannot figue out how to find them :(

I know that there are 3 equivalence classes there: [0], [1], [2], so the numbers should be equivalent to one of these...


Thanks´s very much in advance,
marin

 

[ramsey2879]

Hi there!

I need a little help from your side. Could you give me an example of a quadratic number in Z₃

there is a lemma, that says there are exactly two quadratic numbers there, but I somehow cannot figue out how to find them :(

I know that there are 3 equivalence classes there: [0], [1], [2], so the numbers should be equivalent to one of these...


Thanks´s very much in advance,
marin

I am not familiar with proper terminology, so please excuse if I say this wrong. Z3 is the equivalence class mod 3 so we need to find which values can equal x*x mod 3. The squares 1,4,9,16,25,36 ... are respectively equal to 1,1,0,1,1,0 mod 3
so the two quadratic numbers are 1 and 0 in Z3. To find the quadratic numbers in Z13 you simply take the values of x*x mod 13 as x goes from 0 to 6 since 7 = -6 mod 13 and 7*7 = 6*6 mod 13.

 

[HallsofIvy]

Hi there!

I need a little help from your side. Could you give me an example of a quadratic number in Z₃

there is a lemma, that says there are exactly two quadratic numbers there, but I somehow cannot figue out how to find them :(

I know that there are 3 equivalence classes there: [0], [1], [2], so the numbers should be equivalent to one of these...


Thanks´s very much in advance,
marin

I'm not clear on what you mean by "quadratic numbers". Do you mean just numbers, y, such that x²= x*x= y? If so, then look at 0*0= 0, 1*1= 1, 2*2= 4= 1 mod 3. the two "quadratic numbers", if this is what you mean, in Z₃ are [0] and [1].