# Hensel's lemma

• Nov 11th 2010, 02:23 PM
Chris11
Hensel's lemma
My question is if Hensel's lemma is an if and only if. THis is for polynomial congruences. For instance, if there are no solutions to $x^2=a(mod p)$, do I get to conclude that there are no solutions to $x^2=a(mod p^2)$, and so on?
• Nov 11th 2010, 07:26 PM
tonio
Quote:

Originally Posted by Chris11
My question is if Hensel's lemma is an if and only if. THis is for polynomial congruences. For instance, if there are no solutions to $x^2=a(mod p)$, do I get to conclude that there are no solutions to $x^2=a(mod p^2)$, and so on?

Well, trivially: if $x^2=a\!\!\pmod{p^2}$ , then $x^2=a+kp^2=a+(kp)p\Longrightarrow x^2=a\!\!\pmod p$...(Nerd)

Tonio