A question about quadratic residues

  • Thread starter Thread starter yeland404
  • Start date Start date
  • Tags Tags
    Quadratic
yeland404
Messages
23
Reaction score
0
I need to prove that a be a odd integer that congruence X^2\equiva mod 2
is always solvable with exactly one incongruent solution modulo 2.
this question is linked with (b) let a be an odd integer. Prove that the congruence X^2\equiva mod 4 is solvable iff a\equiv1 mod 4. in this case ,prove that X^2\equiva mod 4solutions has exactly two incongruent
solutions modulo 4.

these two seem to link with each other. And the proposition I learn is X^2\equiva mod p has either no solution or two solutions, but p there is an odd prime number. HOw to apply to the queations above?
 
Physics news on Phys.org
You should be able to just check these by hand. For example does x^2=2 (mod 4) have any solutions? Just plug in 0,1,2,3 for x and see what you get
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

Algorithm to find square root of a quadratic residue mod p
Replies
2
Views
2K
Number of quadratic residues mod N>1
Replies
3
Views
3K
Residues and non residues of general quadratic congruences
Replies
1
Views
2K
MHB Prove the polynomial f(x)=x^2-q is irreducible in F_p[x]?
Replies
1
Views
2K
Is a Quadratic Nonresidue of Odd Primes p and q Also a Nonresidue Modulo pq?
Replies
2
Views
4K
Congruence: Solving Let p be a Prime Number
Replies
6
Views
3K
Order of a mod m & quadratic residues
Replies
3
Views
4K
When n*(n+1)/2 - k is never a square
Replies
2
Views
1K
Where does p = 5 (mod 8) solve x^2 = a?
Replies
5
Views
8K
Quadratic Integers: Understanding the Theorem and Proving 32 = ab in Q[sqrt -1]
Replies
3
Views
3K

Hot Threads

Recent Insights

Back
Top