No Arabic abstract
Let $ksubseteq K$ be a finite Galois extension of fields with Galois group $G$. Let $mathscr{G}$ be the automorphism $k$-group scheme of $K$. We construct a canonical $k$-subgroup scheme $underline{G}subsetmathscr{G}$ with the property that $Spec_k(K)$ is a $k$-torsor for $underline{G}$. $underline{G}$ is a constant $k$-group if and only if $G$ is abelian, in which case $G=underline{G}$.
Quantization is studied from a viewpoint of field extension. If the dynamical fields and their action have a periodicity, the space of wave functions should be algebraically extended `a la Galois, so that it may be consistent with the periodicity. This was pointed out by Y. Nambu three decades ago. Having chosen quantum mechanics (one dimensional field theory), this paper shows that a different Galois extension gives a different quantization scheme. A new scheme of quantization appears when the invariance under Galois group is imposed as a physical state condition. Then, the normalization condition appears as a sum over the product of more than three wave functions, each of which is given for a different root adjoined by the field extension.
In this paper we show an explicit polynomial in Q[x] that has Galois group SL2(F16), filling in a gap in the tables of Juergen Klueners and Gunther Malle. The computation of this polynomial uses modular forms and their Galois representations.
The arboreal Galois group of a polynomial $f$ over a field $K$ encodes the action of Galois on the iterated preimages of a root point $x_0in K$, analogous to the action of Galois on the $ell$-power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 - 1$ when the field $K$ and root point $x_0$ satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of $f$. For $K=mathbb{Q}$, our condition holds for infinitely many choices of $x_0$.
We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $rho$ is determined by an equality of an $m$-power character $gmapsto Tr(rho(g^m))$ for some natural number $m$. Using this, we extend Faltings finiteness criteria to determine the equivalence of two $ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.
For positive integers $n$, the truncated binomial expansions of $(1+x)^n$ which consist of all the terms of degree $le r$ where $1 le r le n-2$ appear always to be irreducible. For fixed $r$ and $n$ sufficiently large, this is known to be the case. We show here that for a fixed positive integer $r e 6$ and $n$ sufficiently large, the Galois group of such a polynomial over the rationals is the symmetric group $S_{r}$. For $r = 6$, we show the number of exceptional $n le N$ for which the Galois group of this polynomial is not $S_r$ is at most $O(log N)$.