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

Thread starter k3232x
Start date Nov 1, 2015
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Polynomial

Nov 1, 2015

k3232x

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].

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Nov 1, 2015

Euge

Gold Member

MHB

POTW Director

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 'About the existence of Hamel basis for vector spaces'

It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC). Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?

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$Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'$

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...

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

Thread 'About the existence of Hamel basis for vector spaces'

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

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