I am looking for examples of families of tamely ramified extensions of $\mathbb{Q}$.


  • $\begingroup$ This can be useful for some readers: math.stackexchange.com/questions/293789/… $\endgroup$ – Piquito Jun 21 '15 at 15:10
  • $\begingroup$ What does tame ramification mean in the context of $\mathbb{Q}$? $\endgroup$ – rondo9 Jun 21 '15 at 16:36
  • $\begingroup$ @rondo: I fear none (assuming of course the possibility of being wrong) $\endgroup$ – Piquito Jun 21 '15 at 17:55
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    $\begingroup$ Fixing an integer $n$, if $f(x)$ is a monic degree $n$ polynomial which is separable (no common factors) modulo $p$ for all primes $p \le n$, then the corresponding extension given by a root of $f(x)$ (or the splitting field) will be tamely ramified everywhere, because it will be unramified at $p \le n$ by construction, and will be tamely ramified at $p > n$ because $p \nmid n!$. For example, if $n = 2$, then any quadratic congruent to $x^2+x+1$ modulo $2$ will work, and if $n = 3$, then any cubic congruent to $x^3+x+1$ modulo $6$ will work. $\endgroup$ – Epargyreus Jun 22 '15 at 0:38
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    $\begingroup$ Another type of example: the cyclotomic fields $\mathbf{Q}(\zeta_N)$ for squarefree $N$. $\endgroup$ – Epargyreus Jun 22 '15 at 0:39

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