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August 1st, 2018, 03:59 AM  #1 
Newbie Joined: Jul 2018 From: morocco Posts: 11 Thanks: 0 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 06:19 AM. 
August 1st, 2018, 05:45 AM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 267 Thanks: 80 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 14) 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 59), 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. 
August 1st, 2018, 07:02 AM  #3 
Newbie Joined: Jul 2018 From: morocco Posts: 11 Thanks: 0 Math Focus: algebraic number theory 
Thank you very much, that was a very rich answer !!!


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condition, fn1, iff, mod, solution 
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