For any finite Galois extension $K$ of $\mathbb Q$ and any conjugacy class $C$ in $\operatorname {Gal}(K/\mathbb Q)$, we show that there exist infinitely many Carmichael numbers composed solely of primes for which the associated class of Frobenius automorphisms is $C$. This result implies that for every natural number $n$ there are infinitely many Carmichael numbers of the form $a^2+nb^2$ with $a,b\in\mathbb Z $.

Let $ E $ be an elliptic curve defined over $\Q,$ of conductor $N$ and without complex multiplication. For any positive integer $l$, let $\phi_l$ be the Galois representation associated to the $l$-division points of~$E$. From a celebrated 1972 result of Serre we know that $\phi_l$ is surjective for any sufficiently large prime $l$. In this paper we find conditional and unconditional upper bounds in terms of $N$ for the primes $l$ for which $\phi_l$ is {\emph{not}} surjective.

For certain real quadratic number fields, we prove density results concerning $4$-ranks of tame kernels. We also discuss a relationship between $4$-ranks of tame kernels and %% $4$-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.

Let $k$ be a cyclic extension of odd prime degree $p$ of the field of rational numbers. If $t$ denotes the number of primes that ramify in $k$, it is known that the Hilbert $p$-class field tower of $k$ is infinite if $t>3+2\sqrt p$. For each $t>2+\sqrt p$, this paper shows that a positive proportion of such fields $k$ have infinite Hilbert $p$-class field towers.