In elementary number theory, we only need to face the problem under the field $\mathbb{C}$. Everything seems self-consistent. So what is the motivation to discuss number theory under the field whose char $\neq $ 0?
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1$\begingroup$ Possible duplicate of Why should I care about fields of positive characteristic? $\endgroup$– Dietrich BurdeJan 26, 2017 at 9:25
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$\begingroup$ @DietrichBurde that question is asking about their need outside of number theory, and takes their use in number theory for granted. $\endgroup$– Mathmo123Jan 26, 2017 at 10:53
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$\begingroup$ Here's a simple application. Whenever we do arithmetic modulo a prime, we are implicitly working in a field of non-zero characteristic. $\endgroup$– Mathmo123Jan 26, 2017 at 10:55
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2 Answers
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Number theory, at it's heart, deals with problems over $\Bbb Z$--and of course more general things that come from motivation from $\Bbb Z$. And one of the most beloved subjects is solving quadratic forms. Thanks to Hensel's Lemma and the celebrated Hasse-Minkowski Theorem we know that solving a given integral quadratic form over $\Bbb Z$ is equivalent to solving it over $\Bbb F_p$ for all odd $p$ and $\Bbb Z/8$ for $p=2$ and over $\Bbb R$. This alone is an enormous number-theoretic reason which would justify studying $\Bbb F_p$ all by itself.
Even in elementary number theory, we use modular arithmetic, which--in the prime case--is arithmetic over $\Bbb F_p$, so I would also dispute the underlying claims you make in your problem statement. Congruences are an enormously important part of study in number theory. Dealing with modular arithmetic integers and Dirichlet characters leads us to a result on $\Bbb Z$ by using $\zeta$-functions to prove Dirichlet's Theoreom on Arithmetic Progressions.
There are many, many more reasons to care, even about problems from elementary theory, but I think these two are some of the most central, the latter is closer to pure questions on modular arithmetic rather than specifically primes, but I think in general it illustrates that rings with non-zero characteristic are both interesting and relevant even in the classical theory.
answered Jan 26, 2017 at 19:20
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For one thing, any finite field has characteristic $p$, $p$ a prime. These fields have numerous applications in cryptography for instance. In number theory specifically, many problems over the integers can be solved by reducing them modulo some prime number(s), which leads naturally to finite fields.
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$\begingroup$ We know $\mathbb{Q}$ has a natural algebraic close field $\mathbb{C}$ above it. But for $F_p$, there is no such property. How to solve this difficulty? $\endgroup$ Jan 26, 2017 at 7:41
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2$\begingroup$ The algebraic closure of $\mathbb{Q}$ is not $\mathbb{C}$, it is a subset of the complex numbers called the algebraic numbers. The finite fields also have algebraic closures. $\endgroup$– Leon SotJan 26, 2017 at 7:43
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$\begingroup$ @LeonSot Although the algebraic closure of Q is not C, we can also do everything in the field C. This makes everything "concretely" since C can be concretely construct in terms of Q (Using Cauchy sequence). For F_p, can we construct a field similar as C? $\endgroup$ Jan 26, 2017 at 7:53
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1$\begingroup$ I'm unsure what you consider to be "concrete", C cannot be constructed from Q with cauchy sequences (R by definition is the completion of Q in this sense.) The finite fields F_p have algebraic closures sitting above them, and in some sense they are quite natural as an increasing union of finite fields. $\endgroup$– Leon SotJan 26, 2017 at 8:06
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1$\begingroup$ Then the algebraic closure of F_p is constructible. $\endgroup$– Leon SotJan 26, 2017 at 10:08