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In the framework of classical motives, there is no such thing as a motive $\mathbb Q(-\tfrac 12)$, i.e. a tensor root of $\mathbb Q(-1)$. There is one, however, in a more general setting of "exponenti …

Let $n>1$ be an integer, and let us consider the set $P(n)$ of all prime numbers $p$ such that $p$ is not congruent to $1$ modulo $n$. Dirichlet's Density Theorem tells us that $P(n)$ has a natural de …

Let $A$ be the set of odd integers $\geq 0$, and let $B$ be the set of those integers $n$ which are odd if $2^m \leq n < 2^{m+1}$ for an odd $m$ and even if $2^m \leq n < 2^{m+1}$ for an odd $m$. Both …

Intuitively, the Galois group (of a splitting field over $\mathbb Q$ of) a polynomial $f\in\mathbb Q[X]$ taken at random is most probably the full permutation group on the roots of $f$. This intuition …

Let $N$ be a nonzero rational number. For every prime number $p$ with $v_p(N)=0$, let $a_p$ denote the index in $\mathbb Z/p\mathbb Z$ of the subgroup generated by $N$ modulo $p$. So we have $a_p=1$ i …

Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category …