Let $h_n$ denote the class number of $n$-th layer of the cyclotomic ${\mathbb{Z}}_2$-extension of $\mathbb{Q}$. Weber proved that $h_n(n\ge 1)$ is odd and Horie proved that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ satisfying $\ell \equiv 3,5(mod8)$. In a previous paper, the authors showed that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ less than ${10}^7$. In this paper, by investigating properties of a special unit more precisely, we show that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ less than $1.2·{10}^8$. Our argument also leads to the conclusion that $h_n(n\ge 1)$ is not divisible by...

For a prime $p$ and positive integers $\ell \<k\<h\<p$ with $d=(h,k,\ell ,p-1)$, we show that $M$, the number of simultaneous solutions $x,y,z,w$ in ${\mathbb{Z}}_p^{*}$ to $x^h+y^h=z^h+w^h$, $x^k+y^k=z^k+w^k$, $x^{\ell }+y^{\ell }=z^{\ell }+w^{\ell }$, satisfies $${\displaystyle M\le 3d^2{(p-1)}^2+25hk\ell (p-1).}$$ When $hk\ell =o\left(p{d}^2\right)$ we obtain a precise asymptotic count on $M$. This leads to the new twisted exponential sum bound $${\displaystyle \left|\sum _{x=1}^{p-1}\chi \left(x\right)e^{2\pi if\left(x\right)/p}\right|\le 3^{\frac{1}{4}}{d}^{\frac{1}{2}}{p}^{\frac{7}{8}}+\sqrt{5}{\left(hk\ell \right)}^{\frac{1}{4}}{p}^{\frac{5}{8}},}$$ for trinomials $f=a{x}^h+b{x}^k+c{x}^{\ell }$, and to results on the average size of such sums.

Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $\left(H_{p^n}^{\prime }\right)$ when any abelian extension $N/F$ of exponent dividing $p^n$ has a normal integral basis with respect to the ring of $p$-integers. We also say that $F$ satisfies $\left(H_{p^{\infty }}^{\prime }\right)$ when it satisfies $\left(H_{p^n}^{\prime }\right)$ for all $n\ge 1$. It is known that the rationals $\mathbb{Q}$ satisfy $\left(H_{p^{\infty }}^{\prime }\right)$ for all prime numbers $p$. In this paper, we give a simple condition for a number field $F$ to satisfy $\left(H_{p^n}^{\prime }\right)$ in terms of the ideal class group of $K=F\left({\zeta }_{p^n}\right)$ and a “Stickelberger ideal” associated to the...

Let $K$ be a field of degree $n$ over ${\mathbf{Q}}_p$, the field of rational $p$-adic numbers, say with residue degree $f$, ramification index $e$ and differential exponent $d$. Let $O$ be the ring of integers of $K$ and $\mathit{P}$ its unique prime ideal. The trace and norm maps for $K/{\mathbf{Q}}_p$ are denoted $Tr$ and $N$, respectively. Fix $q=p^r$, a power of a prime $p$, and let $\eta$ be a numerical character defined modulo $q$ and of order $o\left(\eta \right)$. The character $\eta$ extends to the ring of $p$-adic integers ${\mathbb{Z}}_p$ in the natural way; namely $\eta \left(u\right)=\eta \left(\tilde{u}\right)$, where $\tilde{u}$ denotes the residue...

Let K, L be algebraic number fields with K ⊆ L, and $O_K$, $O_L$ their respective rings of integers. We consider the trace map $T=T_{L/K}:L\to K$ and the $O_K$-ideal $T\left(O_L\right)\subseteq O_K$. By I(L/K) we denote the group indexof $T\left(O_L\right)$ in $O_K$ (i.e., the norm of $T\left(O_L\right)$ over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of $T\left(O_L\right)$ (Theorem 1)....