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A special case of Cheboratev's density theorem states that, for $K/\mathbb{Q}$ a Galois number field of degree $n$, then the rational primes that split completely in $K$ have density $1/n$.

Is there an elementary proof of this fact?

Actually I'm asking this only to apply it to $K=\mathbb{Q}(\zeta_n)$ and get that the density is nonzero, and there is an elementary proof of this. However I'm also interested in the more general result.

GH from MO
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Meow
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    Should this be called Chebotarev's theorem? It is after all a rather more basic 19th century theorem, known to Kronecker and Frobenius (with the logarithmic notion of density), and also the Galois $K/\mathbb{Q}$ case (with the natural notion of density) of Landau's prime ideal theorem. Chebotarev's contribution treating the non-identity conjugacy classes in the Galois group was the difficult new step that he solved at the time, leading Artin to the proof of his conjectured abelian reciprocity law. – Vesselin Dimitrov Sep 27 '15 at 05:19
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    With the logarithmic notion of density being understood, this case has fully elementary proofs by Chebyshev ideas. I have sketched them here: http://mathoverflow.net/questions/25794/shortest-most-elegant-proof-for-l1-chi-neq-0/194795#194795 and here: http://mathoverflow.net/questions/208604/about-the-prime-divisors-of-values-of-polynomials/208611#208611 . – Vesselin Dimitrov Sep 27 '15 at 05:21
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    You could have a look to THEOREM 3.4 in Milne's Class Field Theory notes (p. 188). – Watson Dec 22 '16 at 08:52

1 Answers1

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There does exist an elementary proof, and is given in many books on algebraic number theory. I think it is in Lang's book. I reproduce the proof below.

Consider $ \zeta _K(s)=\prod _{\mathfrak p}(1-\frac{1}{(N\mathfrak p)^{s}})^{-1}$ where $\mathfrak p$ runs over all prime ideals in the ring of integers in $K$. Therefore,

$$\log (\zeta _K(s))=\sum _{\mathfrak p}\sum _ {m\geq 1} \frac{1}{m(N\mathfrak p)^{ms}}$$

Since $\zeta _K(s)=\frac{1}{s-1}(a_0+a_1(s-1)+\cdots)$ with $a_0\neq 0$, it follows that $\frac{\log (\zeta _K(s))}{\log \frac{1}{s-1}}$ tends to $1$ as $s$ tends to $1$ from the right. On the other hand, in this limit, only the term $m=1$ and $\mathfrak p$ of degree one over $\mathbb Q$ need be considered. Hence we get $$1= \lim _{s \rightarrow 1} \frac{\sum _{\mathfrak p} \frac{1}{(N\mathfrak p) ^s}}{\log \frac{1}{s-1}}.$$ Now, degree $1$ primes $\mathfrak p$ lie over rational primes $p$ which split completely in $K$, and over each $p$, there are $n=\deg(K/{\mathbb Q})$ primes $\mathfrak p$ of degree $1$. Hence we get

$$1=\lim _{s\rightarrow 1} n\left( \frac{\sum _p \frac{1}{p^s}}{\log \frac{1}{s-1}}\right)$$ where the sum is over primes $p$ which split completely. This gives what you want.

Myshkin
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Venkataramana
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  • I've added \ to log, lim and deg in the TeXt, I hope you don't mind :) – M.G. Sep 20 '15 at 12:38
  • Thank you. Of course I don't mind, and it will help me in future too! – Venkataramana Sep 20 '15 at 12:38
  • Why does this proof only work when K/Q is Galois? What would be the main difficulty in extending it? – Campello Oct 26 '15 at 16:57
  • If $K/\mathbb Q$ is not Galoisian, then, degree one primes $\mathfrak p$ in $K$ do not correspond to primes in $\mathbb Q$ which split completely in $K$. So the above count does not work – Venkataramana Oct 27 '15 at 00:03
  • But then you can consider the closure of K :) – Campello Oct 27 '15 at 09:20
  • to do what? some primes in $K$ are of degree one and some not, so the above count does not work for non Galoisian.. – Venkataramana Oct 27 '15 at 11:21
  • @Myshkin: thank you for the edit. I did not know how to put the big bracket. – Venkataramana Oct 19 '16 at 06:19
  • @Venkataramana Use \left( and \right) for putting big brackets. BTW, though i don't understand a thing in this answer, i wanted to ask : Are these $\zeta_{K}(s)$ connected to $L$ functions in any sense. Just by looking at your answer i see similar ideas used in the proof Dirichlets theorem for primes in Arithmetic Progression :) – C.S. Jun 19 '18 at 10:20
  • @crskhr: yes, the $\zeta _K(s)$ is the $L$ function associated to the trivial character. – Venkataramana Jun 19 '18 at 11:56
  • @Venkataramana Thanks for taking time to respond :) – C.S. Jun 19 '18 at 12:20
  • @crskhr: Not at all! Thanks for taking time to write. – Venkataramana Jun 19 '18 at 14:57
  • Since this uses the fact that $\zeta_K(s)$ has a simple pole at $1$, it can't be called an 'elementary' proof. It is more accurate to say that it is simpler than the general Chebotarev density theorem. – A Stasinski Oct 08 '21 at 14:50