MATHEMATICS

B. M. URAZBAEV

ON THE GROWTH OF THE NUMBER OF COMPLETELY CRITICAL CYCLIC FIELDS OF DEGREE \(l^h\)

(Presented by Academician I. M. Vinogradov on 19 XI 1956)

This note considers the question of the distribution of completely critical cyclic fields of degree \(l^h\), whose discriminants do not exceed a given bound, where \(l\) is a prime\(^*\); \(h\) is any positive integer. A field \(K\) will be called completely critical if all prime divisors of the discriminant of \(K\) are completely critical prime numbers of the field \(K\).

There exists, obviously, an infinite set of completely critical fields. For example, even the number of completely critical cyclic fields of degree \(l^h\) whose discriminants consist of only one prime divisor is infinite, since there are infinitely many prime numbers of the form \(p \equiv 1 \pmod {l^h}\). In particular, the completely critical fields include cyclic fields having degree \(l\).

Relying on group-theoretic considerations and on the Kronecker–Weber theorem on abelian fields over the field of rational numbers, one can derive a general expression for the discriminant of abelian fields of degree \(l^\alpha\), \(\alpha\) an integer.

Let \(K_1\) be an abelian field of degree \(l^\alpha\) over the field of rational numbers; \(G\) the Galois group of the field \(K_1\). Let \(G\) be a group of type

\[ (l^h,\ldots,l^h;\ l^{h_1},\ldots,l^{h_1},\ldots,l^{h_\nu},\ldots,l^{h_\nu}), \tag{1} \]

where the component \(l^h\) occurs \(k\) times, \(l^{h_1}\)—\(k_1\) times, and so on, \(l^{h_\nu}\)—\(k_\nu\) times,

\[ kh+k_1h_1+\cdots+k_\nu h_\nu=\alpha;\qquad h>h_1>\cdots>h_\nu\geq 1. \]

Theorem 1. The discriminant \(D\) of an abelian field \(K_1\) of type (1) is represented in the form

\[ D=\left(\prod_{i_1} p_{i_1}^{(h)}\right)^{H\left(1-\frac{1}{l^h}\right)} \left(\prod_{i_2} p_{i_2}^{(h-1)}\right)^{H\left(1-\frac{1}{l^{h-1}}\right)} \cdots \left(\prod_{i_h} p_{i_h}^{(1)}\right)^{H\left(1-\frac{1}{l}\right)}, \]

where \(H=l^\alpha\); \(p_{i_\mu}^{(h+1-\mu)}\) are distinct prime numbers of the form \(1+tl^{h+1-\mu}\) \((\mu=1,2,\ldots,h)\); \(i_\mu\geq 0\), but \(i_1\geq 1\).

As a consequence of this theorem, one obtains the discriminants of the fields that were considered in the work \((^2)\).

Theorem 2. The discriminant of a completely critical abelian field of degree \(l^\alpha\) has the form

\[ (p_1p_2\cdots p_m)^{H\left(1-\frac{1}{l^h}\right)}, \]

where \(H=l^\alpha\); \(p_i\equiv 1 \pmod {l^h}\) \((i=1,2,\ldots,m)\) are distinct prime numbers, \(m\geq 1\).

\(^*\) It is assumed that \(l\) is a noncritical prime number.

Theorem 3. The discriminant of a completely critical cyclic field of degree \(l^h\) is

\[ (p_1p_2\cdots p_m)^{l^h-1}, \tag{2} \]

where \(p_i\equiv 1\pmod {l^h}\) \((i=1,2,\ldots,m)\) are distinct prime numbers, \(m\geqslant 1\).

Theorem 4. The number of completely critical cyclic fields of degree \(l^h\) having discriminant (2) is equal to \(\varphi(l^h)^{m-1}\).

For the proof, see paper \((^3)\).

Theorem 5. The number \(N\) of completely critical cyclic fields of degree \(l^h\) whose discriminants do not exceed \(x^{l^h-1}\) is given by the asymptotic formula

\[ N=\lambda x+O\left(x^{1-\frac{1}{\varphi(l^h)}+\varepsilon}\right), \tag{3} \]

where \(\lambda>0\) is a constant depending only on the structural properties of the fields under consideration; \(\varepsilon>0\) is an arbitrary number.

The proof, as in paper \((^2)\), is carried out by the classical method of the distribution of prime numbers, using results from the theory of Dirichlet \(L\)-functions \((^1)\). The function considered for this purpose,

\[ f(s)=\prod_{p\equiv 1(l^h)}\left(1+\frac{\varphi(l^h)}{p^s}\right),\qquad s=\sigma+it, \]

is regular in the half-plane \(\sigma>1/2\) and has there a single pole of first order at \(s=1\), since in the representation

\[ f(s)=\prod_{\chi} L(s;\chi)\cdot \prod_{p\equiv 1(l^h)} \left(1-\frac{1}{p^s}\right)^{\varphi(l^h)} \left(1+\frac{\varphi(l^h)}{p^s}\right)\cdot \prod_{p\not\equiv 1(l^h)} \left(1-\frac{1}{p^{fs}}\right)^g \]

the function defined by the first product on the right is regular in the whole plane of complex numbers, except for a simple pole at \(s=1\), while the last two products converge absolutely and uniformly for \(\sigma>1/2\). Here \(\chi\) runs through all Dirichlet characters \((\bmod\, l^h)\); \(L(s;\chi)\) is the Dirichlet function of the character \(\chi\); \(fg=\varphi(l^h)\); \(f\) is the exponent to which \(p\) belongs \((\bmod\, l^h)\).

The constant \(\lambda\) in formula (3) is represented by the expression

\[ \lambda=\frac12\left(1-\frac1l\right) \prod_{p\equiv 1(l^h)} \left(1-\frac1p\right)^{\varphi(l^h)} \left(1+\frac{\varphi(l^h)}p\right)\cdot \prod_{p\not\equiv 1(l^h)} \left(1-\frac{1}{p^f}\right)^g \lim_{s\to 1}\prod_{\chi\ne\chi_0} L(s;\chi), \]

which, by Dirichlet’s theorem on the distribution of prime numbers in progressions, is nonzero; moreover, \(\lambda x\) is the residue of the function

\[ \frac{f(s)}{s(s+1)}x^s \]

at the point \(s=1\), integrated according to the Riemann–Hadamard construction.

Alma-Ata State
Pedagogical Institute named after Abai

Received
12 XI 1956

REFERENCES

\(^{1}\) N. G. Chudakov, Introduction to the Theory of Dirichlet \(L\)-Functions, 1947.
\(^{2}\) B. M. Urazbaev, DAN, 95, No. 5, 935 (1954); 105, No. 4, 659 (1955).
\(^{3}\) B. M. Urazbaev, Izv. AN KazSSR, ser. astr., phys., math. and mech., issue 3 (7), 51 (1953).