P-adic Number help please

May 2010
2
0
London
Hi (Nod)

I was wondering if anyone could help me on this problem I am having.

I have the equation:

y^2 = 2( x^4 -17)
(which I am showing has no real solutions but has solutions in the p-adic numbers i.e. a counterexample to the hasse principle)

I need to show that it has a non trivial solution in R and in Q_p (p-adic numbers for p=2, 17 (the 2-adics and 17-adic)

Any help or advice would be so much apprciated! I`m completely stumped!

THANKS!!

Cara

xxxx
 
Last edited by a moderator:

chiph588@

MHF Hall of Honor
Sep 2008
1,163
429
Champaign, Illinois
What do you mean by no trivial solution in \(\displaystyle \mathbb{R} \)?

In \(\displaystyle \mathbb{R} \) we have solutions \(\displaystyle \left(x,\pm\sqrt{2(x^4-17)}\right) \) for all \(\displaystyle |x|>\sqrt[4]{17} \).
 
May 2010
2
0
London
Sorry that was a typo! Thats meant to be in Q.
this equation has nontrivial solutions over​
Qp for all prime numbers p and over R but has no nontrivial solutions over Q.
I need to show that although there is no solution in Q there is a solution in Qp . But I don`t know how to show that for the case where modulo p=2 and modilo p=17?

Its a special case, but its completely confused me.
Thanks so much for the reply!