For all primes p, show that is solvable.

From a theorem, the question is equivalent to showing that

And then I am stuck.(Worried)

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- November 14th 2009, 12:08 PMcomssaprove that a congruence is solvable
For all primes p, show that is solvable.

From a theorem, the question is equivalent to showing that

And then I am stuck.(Worried) - November 14th 2009, 03:45 PMchiph588@
There are four options for since is the exhaustive list of factors of .

If , it's pretty easy to see .

If .

If .

If .

In this case, and hence is a square modulo , i.e. . Therefore we can choose such that . So .

Therefore is always solvable.