Open MathsAll courses

Uniform complete splitting in Kummer fields

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This lesson proves Proposition 5.2 of Artin's conjecture and the index of a prime: the number of primes \(p\in(x,2x]\) that split completely in the Kummer field \(K_q=\mathbb Q(\mu_q,a^{1/q})\) is \(O\bigl(x/(q(q-1)\log x)+x^{1-\delta_0}\bigr)\), uniformly for primes \(q\le\exp((\log x)^{0.3})\). The field \(K_q\) is not cyclotomic, but after adjoining \(\mu_{12q}\) it becomes an abelian extension of the cyclotomic field \(\mathbb Q(\mu_{12q})\), to which the zero-free strip of The zero-free strip applies; the strip then descends to \(\zeta_{K_q}\). A smoothed explicit formula, whose error depends on the field only through its degree and discriminant, converts the zero-free strip into the count. The uniformity in \(q\) rests on a zero count for Dedekind zeta functions with absolute constants, used as a named result.

Prerequisites. Artin's conjecture and the index of a prime, Section 3 and Proposition 5.2; The zero-free strip, Corollary 4.2; Cyclotomic fields, Theorem 12.2; The different and the discriminant, (1)–(4), Proposition 14.2 and Theorem 14.3; Discriminants and integral bases, Proposition 2.2; Hilbert's theorem 90 and Kummer theory, Theorem 3.3; Artin L-functions, conductors and discriminants, Proposition 21.1 and Theorem 21.2; Explicit formulas for Dirichlet L-functions and primes in progressions, Section 6 and Theorem 6.1; The Gamma function and Stirling's formula, Section 5.

1. Degree, discriminant and the zero-free strip

Fix an integer \(a\) with \(|a|>1\), and recall \(E_q=\mathbb Q(\mu_q)\), \(K_q=\mathbb Q(\mu_q,a^{1/q})\) from Lesson 1. Put \(\delta_0=10^{-6}\).

Lemma 1.1 (the Kummer fields [OpenAI-029, Lemma 9.1]). For every sufficiently large prime \(q\), the degree \(n_q=[K_q:\mathbb Q]\) and the absolute discriminant \(D_q\) satisfy \[ \begin{gathered} n_q=q(q-1),\\ D_q\le q^{q(q-2)}\bigl(q^q|a|^{q-1}\bigr)^{q-1},\\ \log D_q+n_q\le C_a\,n_q\log(2q),\quad(1.1) \end{gathered} \] and \[ \begin{gathered} \zeta_{K_q}(s)\ne0\\ (\operatorname{Re}s>1-\delta_0).\quad(1.2) \end{gathered} \]

Proof. Degree. Choose a prime \(\ell\mid a\) and take \(q\ne\ell\), \(q>v_\ell(a)\). By Theorem 12.2 of Cyclotomic fields, \(|d_{E_q}|=q^{q-2}\), so \(\ell\) is unramified in \(E_q\) and every prime of \(E_q\) above \(\ell\) gives \(a\) the valuation \(v_\ell(a)\), which is not divisible by \(q\). Hence \(a\) is not a \(q\)th power in \(E_q\). Since \(\mu_q\subset E_q\), Kummer theory (Theorem 3.3 of Hilbert's theorem 90 and Kummer theory) makes \(K_q/E_q\) cyclic, with \(\sigma\mapsto\sigma(a^{1/q})/a^{1/q}\) embedding its Galois group in \(\mu_q\); the degree divides \(q\) and is not \(1\), so it is \(q\), and \(n_q=q(q-1)\).

Discriminant. Let \(\alpha=a^{1/q}\), with minimal polynomial \(f(T)=T^q-a\) over \(E_q\), and let \(B\) be the ring of integers of \(K_q\), \(A\) that of \(E_q\). The trace-dual of \(A[\alpha]\) is \(f'(\alpha)^{-1}A[\alpha]\) (Euler's lemma, in the proof of Proposition 14.2 of The different and the discriminant), and it contains the trace-dual \(J_{B/A}\) of the larger ring \(B\), by (1) there. Hence \(f'(\alpha)J_{B/A}\subseteq A[\alpha]\subseteq B\), that is, \(f'(\alpha)\in\mathfrak D_{B/A}\) by (4) there. Taking norms with (15) of Theorem 14.3, the relative discriminant divides \(N_{K_q/E_q}(f'(\alpha))A\), and by Proposition 2.2 of Discriminants and integral bases this is the discriminant of \(f\), \(\pm q^qa^{q-1}\). The tower formula (16) of Theorem 14.3 then gives \[ \begin{aligned} D_q&=|d_{E_q}|^{q}\,\operatorname N_{E_q/\mathbb Q}(\mathfrak d_{K_q/E_q})\\ &\le q^{q(q-2)}\bigl(q^q|a|^{q-1}\bigr)^{q-1}, \end{aligned} \] and taking logarithms gives the last bound in (1.1).

The zero-free strip. Put \(F_q=\mathbb Q(\mu_{12q})\), a cyclotomic field containing \(\mu_{12}\), and \(\widetilde K_q=K_qF_q\). The extension \(\widetilde K_q/F_q\) is cyclic, being generated by \(a^{1/q}\) over a field containing \(\mu_q\). The extension \(\widetilde K_q/K_q\) is abelian, since restriction embeds its Galois group in the abelian group \(\operatorname{Gal}(F_q/\mathbb Q)\). By (4) of Proposition 21.1 and Theorem 21.2 of Artin L-functions, conductors and discriminants, for an abelian extension the Dedekind zeta function of the top field is the product of the primitive Hecke \(L\)-functions of the characters of its Galois group: \[ \begin{gathered} \zeta_{\widetilde K_q}(s)=\prod_{\eta}L_{F_q}(s,\eta),\\ \frac{\zeta_{\widetilde K_q}(s)}{\zeta_{K_q}(s)}=\prod_{\eta'\ne1}L_{K_q}(s,\eta'), \end{gathered} \] with \(\eta\) running over the characters of \(\operatorname{Gal}(\widetilde K_q/F_q)\) and \(\eta'\) over the nontrivial characters of \(\operatorname{Gal}(\widetilde K_q/K_q)\), viewed as finite-order Hecke characters. Every \(L_{F_q}(s,\eta)\) has no zero in \(\operatorname{Re}s>1-\delta_0\) by Corollary 4.2 of Lesson 14, so neither has \(\zeta_{\widetilde K_q}\). Every \(L_{K_q}(s,\eta')\) is entire, \(\eta'\) being nonprincipal. A zero of \(\zeta_{K_q}\) in the half-plane would therefore be a zero of \(\zeta_{\widetilde K_q}\). \(\square\)

The width \(\delta_0\) does not depend on \(q\); everything else that enters below depends on \(K_q\) only through \(n_q\) and \(D_q\).

2. A uniform zero count

For a number field \(K\) of degree \(n_K\) and absolute discriminant \(d_K\), let \(N_K(T)\) be the number of zeros \(\rho=\beta+i\gamma\) of \(\zeta_K\) with \(0<\beta<1\) and \(|\gamma|\le T\), counted with multiplicity.

Named result N4 (Hasanalizade, Shen and Wong [HSW, Corollary 1.2]). For every number field \(K\) and every \(T\ge1\), \[ \begin{aligned} &\Bigl|N_K(T)-\frac T\pi\log\Bigl(d_K\Bigl(\frac T{2\pi e}\Bigr)^{n_K}\Bigr)\Bigr|\\ &\quad \le0.228\,(\log d_K+n_K\log T)\\ &\qquad +23.108\,n_K+4.520 . \end{aligned} \] It is used without proof in the programme.

Corollary 2.1. There is an absolute constant \(C\) such that for every number field \(K\) and every real \(t\), \[ \begin{aligned} &\#\{\rho:\ t\le\gamma\le t+1\}\\ &\le C\bigl(\log d_K+n_K\log(2+|t|)\bigr).\quad(2.1) \end{aligned} \]

Proof. For \(|t|\ge2\) the zeros counted lie in \(|t|-1\le|\gamma|\le|t|+1\), so their number is at most \(N_K(|t|+1)-N_K(|t|-1-\epsilon)\) for small \(\epsilon>0\). By N4 the main terms differ by at most \(\frac2\pi\log(d_K((|t|+2)/(2\pi))^{n_K})+O(1)\), and the errors add at most twice the right side of N4 at height \(|t|+1\). For \(|t|<2\) use \(N_K(3)\). Both are at most \(C(\log d_K+n_K\log(2+|t|))\). \(\square\)

3. A smooth prime-ideal estimate

Lemma 3.1 (smooth prime ideals [OpenAI-029, Lemma 9.2]). Fix \(0<\delta<\frac12\) and a nonnegative \(\phi\in C_c^\infty([\frac12,3])\). Let \(K\) be a number field of degree \(n\) and absolute discriminant \(D\) such that \(\zeta_K\) has no zero in \(\operatorname{Re}s>1-\delta\). For every \(x\ge2\), \[ \begin{aligned} &\Theta_{K,\phi}(x)\\ &\quad=\sum_{\mathfrak p}\sum_{m\ge1}(\log\operatorname N\mathfrak p)\,\phi\Bigl(\frac{\operatorname N\mathfrak p^m}x\Bigr)\\ &\quad\le C_{\phi}\bigl(x+x^{1-\delta}(\log D+n)\bigr),\quad(3.1) \end{aligned} \] with \(C_\phi\) independent of \(K\) and \(x\); the sum runs over the nonzero prime ideals of \(K\).

Proof. Apply Theorem 6.1 of Explicit formulas for Dirichlet L-functions and primes in progressions to the trivial Hecke character of \(K\), for which \(L=\zeta_K\), \(Q=D\), the completed function has its two poles at \(0\) and \(1\), and \(G(s)=\Gamma_{\mathbb R}(s)^{r_1}\Gamma_{\mathbb C}(s)^{r_2}\), with the test function \(f(y)=\phi(y/x)\). Its Mellin transform is \(H(s)=\int_0^\infty f(y)y^s\,dy/y=x^s\Phi(s)\), with \(\Phi(s)=\int_0^\infty\phi(t)t^s\,dt/t\) entire and, by repeated integration by parts, \(|\Phi(\sigma+it)|\le C_{\phi,A}(1+|t|)^{-A}\) on \(0\le\sigma\le1\). Since \(x\ge2\), \(f(1)=0\), and \(f^\sharp(y)=y^{-1}f(y^{-1})\) vanishes for \(y\ge1\); so the dual prime sum and the conductor term disappear, and (6.3) there reads \[ \begin{aligned} &\Theta_{K,\phi}(x)=H(1)+H(0)-\sum_\rho H(\rho)\\ &\quad+\frac1{2\pi}\int_{\mathbb R}2\operatorname{Re}\frac{G'}G\bigl(\tfrac12+it\bigr)\\ &\qquad\times H\bigl(\tfrac12+it\bigr)\,dt , \end{aligned} \] the sum running over the zeros of the completed function, which are the zeros of \(\zeta_K\) with \(0<\operatorname{Re}\rho<1\).

Here \(H(1)=x\Phi(1)\) and \(H(0)=\Phi(0)\). Every zero has \(\operatorname{Re}\rho\le1-\delta\), so \(|H(\rho)|\le x^{1-\delta}C_{\phi,3}(1+|\gamma|)^{-3}\); grouping the zeros in unit intervals of height and using (2.1), \[ \begin{aligned} &\sum_\rho|H(\rho)|\le C\,x^{1-\delta}\sum_{k\in\mathbb Z}(1+|k|)^{-3}\\ &\qquad\times\bigl(\log D+n\log(2+|k|)\bigr)\\ &\quad\le C\,x^{1-\delta}(\log D+n). \end{aligned} \] Finally \(\frac{G'}G\) is a sum of \(r_1+r_2\le n\) logarithmic derivatives of \(\Gamma_{\mathbb R}\) and \(\Gamma_{\mathbb C}\), each \(O(\log(2+|t|))\) on \(\operatorname{Re}s=\frac12\) (Section 5 of The Gamma function and Stirling's formula); so the integral is at most \(C\,x^{1/2}n\)\(\int(\log(2+|t|))|\Phi(\frac12+it)|\,dt\)\(\le C_\phi x^{1/2}n\le C_\phi x^{1-\delta}n\). \(\square\)

Only the first-power terms of prime ideals of degree one matter in the application, but the estimate keeps all terms: they are nonnegative.

4. Proof of the uniform splitting bound

Proof of Proposition 5.2 of Lesson 1. Fix a nonnegative \(\phi\in C_c^\infty([\frac12,3])\) with \(\phi\ge1\) on \([1,2]\). Let \(q\) be large enough for Lemma 1.1, and apply Lemma 3.1 to \(K_q\) with \(\delta=\delta_0\), which is allowed by (1.2). A rational prime \(p\in(x,2x]\) with \(p\nmid aq\) that splits completely in \(K_q\) has \(n_q\) prime ideals above it, each of norm \(p\), and their first-power terms contribute at least \(n_q\log p\ge n_q\log x\) to the nonnegative sum \(\Theta_{K_q,\phi}(x)\). The primes dividing \(aq\) are \(O_a(1)+1\) in number. With \(L=\log x\), (3.1) and (1.1) give \[ \begin{aligned} &\#\{p\ \text{prime}:x<p\le2x,\\ &\qquad p\ \text{splits completely in}\ K_q\}\\ &\le\frac{C_\phi}{n_qL}\bigl(x+x^{1-\delta_0}(\log D_q+n_q)\bigr)+C_a\\ &\le C_a'\Bigl(\frac x{q(q-1)L}+\frac{x^{1-\delta_0}\log(2q)}L\Bigr)\\ &\quad+C_a . \end{aligned} \] For \(q\le\exp(L^{0.3})\), \(\log(2q)/L\le2L^{-0.7}\) is bounded once \(x\) is large, which gives (5.2) of Lesson 1, with constants and a threshold for \(x\) independent of \(q\). \(\square\)

The only property of the cyclotomic fields \(\mathbb Q(\mu_{12q})\) used is the zero-free strip of Lesson 14 with a width independent of \(q\). A width shrinking like \(1/\log q\), as classical zero-free regions give, would only control \(q\) up to a power of \(\log x\) in this argument; this is why Theorem 5.3 of Lesson 1 is stated uniformly over all cyclotomic fields containing \(\mu_{12}\).

5. Exercises

5.1. Compute \(D_q\) exactly for \(a=2\) and \(q=3\), and compare with the bound (1.1).

5.2. Show that \(a\) is a \(q\)th power in \(E_q\) for no prime \(q\) larger than every exponent in the factorization of \(a\), using only the argument of Lemma 1.1.

5.3. In Corollary 2.1, explain why zeros with \(\gamma\) in \([t,t+1]\) satisfy \(|t|-1\le|\gamma|\le|t|+1\), and why the main terms of N4 at heights \(T\) and \(T+2\) differ by \(O(\log d_K+n_K\log(2+T))\).

5.4. Show that \(\log D_q\le q(q-2)\log q\)\(+(q-1)\bigl(q\log q+(q-1)\log|a|\bigr)\) and deduce \(\log D_q\le C_a\,q^2\log q\).

5.5. Suppose a zero-free region of width \(c/\log D_q\) were all that is known for \(\zeta_{K_q}\). Show that the error term \(x^{1-\delta}\log D_q\) would then be smaller than \(x/(q^2\log x)\) only for \(q\) up to a power of \(\log x\).

6. Solutions

5.1. For \(q=3\) the field \(K_3=\mathbb Q(\mu_3,2^{1/3})\) has degree \(6\), and \(D_3=2^4\cdot3^7=34992\) (the splitting field of \(T^3-2\)). The bound is \(3^{3}(3^3\cdot2^2)^2=27\cdot11664=314928\).

5.2. If \(\ell\mid a\) and \(q\) exceeds \(v_\ell(a)\), the valuation of \(a\) at a prime of \(E_q\) above \(\ell\) is \(v_\ell(a)\) for \(q\ne\ell\), which is not divisible by \(q\); a \(q\)th power has valuations divisible by \(q\).

5.3. If \(t\le\gamma\le t+1\) then \(|\gamma|\) lies between \(|t|-1\) and \(|t|+1\). The derivative of \(\frac T\pi\log(d_K(T/(2\pi e))^{n_K})\) in \(T\) is \(\frac1\pi(\log d_K+n_K\log(T/(2\pi)))\), so over an interval of length \(2\) the change is \(O(\log d_K+n_K\log(2+T))\).

5.4. From (1.1), \(\log D_q\le q(q-2)\log q\)\(+(q-1)(q\log q+(q-1)\log|a|)\), which is the stated expression and at most \((2q^2)\log q+q^2\log|a|\le C_aq^2\log q\).

5.5. With \(\delta=c/\log D_q\asymp1/(q^2\log q)\), \(x^{-\delta}=\exp(-c\log x/\log D_q)\) is small only when \(\log D_q\le\epsilon\log x\), that is, \(q^2\log q\ll\log x\).

Sources