MHB Prove the polynomial f(x)=x^2-q is irreducible in F_p[x]?

  • Thread starter Thread starter k3232x
  • Start date Start date
  • Tags Tags
    Polynomial
k3232x
Messages
7
Reaction score
0
If p and q are prime numbers such that p is not a quadratic residue mod q. Show that if pq=-1 mod 4 then the polynomial f(x)=x^2-q is irreducible in F_p[x].
 
Physics news on Phys.org
Hi k3232x,

Welcome. (Wave) Please show what work you've done or what your thoughts are about this problem. That way, we can assist you better.
 

Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
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

I How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
Replies
48
Views
3K
I Irreducible polynomials and prime elements
Replies
5
Views
2K
I Finite fields, irreducible polynomial and minimal polynomial theorem
Replies
6
Views
3K
MHB The polynomial is irreducible over Q(i)
Replies
16
Views
3K
I Equivalent definitons for primitive polynomials
Replies
24
Views
347
I Localizing single variable quotient polynomial ring at a prime ideal
Replies
6
Views
688
I Proof for subgroup -- How prove it is a subgroup of Z^m?
Replies
11
Views
2K
MHB Eisenstein polynomial and field extension
Replies
24
Views
4K
I Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
Replies
0
Views
2K
I Generators of Galois group of ## X^n - \theta ##
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top