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August 1st, 2018, 04:59 AM   #1
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Math Focus: algebraic number theory
f(n)=-1 mod p has a solution iff a condition on p

Hello

Can you please share with me all the results that you know like this:

f(n)=-1 mod p has a solution iff a condition on p

example :
Lagrange Lemma:

x^2=-1 mod p has a solution if and only if p=1 mod 4

1) Is this kind of equation important?
2) I wish that you share with me any book or article that is talking about these arithmetical equations.

Thank you
Best regards

Last edited by skipjack; August 1st, 2018 at 07:19 AM.
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August 1st, 2018, 06:45 AM   #2
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Math Focus: Algebraic Number Theory, Arithmetic Geometry
This sort of question forms a very deep and interesting part of number theory. Let $f$ be polynomial with integer coefficients. You might ask whether there are congruence conditions on $p$ that determine whether $f$ has a root mod $p$. A consequence of class field theory is that such conditions exist precisely when the Galois group of $f$ over $\mathbb{Q}$ is abelian. So, for example, if $f$ is quadratic, then there are always such conditions! And if $f$ is cubic, then there are such conditions precisely when the discriminant of $f$ is a square.

A great starting point that I'd highly, highly recommend is David Cox's book "Primes of the Form $x^2 + ny^2$". Even though the primary aim of the book is to deal with a slightly different problem to yours, it gives a very accessible account of the relevant number theory. Also, your problem does end up getting quite a lot of treatment along the way. For example, one elementary result you'll see is that quadratic reciprocity is equivalent to the following: given odd primes $p \neq q$, the equation $x^2 \equiv q \bmod p$ is solvable if and only if $p \equiv \pm a \bmod 4q$ for some odd integer $a$.

The first section (chapters 1-4) is accessible without much mathematical knowledge; I think being familiar with the language of group theory would help, but you could definitely get by without it and just pick up the necessary ideas along the way. There's a bit of a jump in the middle section (chapters 5-9), as it starts to deal with class field theory. You'll need to be comfortable with Galois theory for this, and you'd probably want to know some algebraic number theory (e.g. ramification + splitting in extensions of number fields, fractional ideals, etc.) to get the most out of it. I personally haven't read the final section - I'd already covered modular forms, complex multiplication, etc. elsewhere, so I can't comment on precisely what you'd need to know to follow his presentation of the material. You'll certainly need a fair amount of complex analysis, in any case.

As well as being a fantastic book in its own right, it also has a great selection of references - if there's any direction you are particularly interested in exploring, you'll be able to go that way.
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August 1st, 2018, 08:02 AM   #3
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Thank you very much, that was a very rich answer !!!
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