Related papers: A simple proof of the Abel-Ruffini theorem

We give a short proof of Chevalley's theorem that every algebraic group is an extension of an Abelian variety by a linear algebraic group. Along the way we treat Bertini's irreducibility theorem.

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier…

We prove that arboreal Galois extensions of number fields are never abelian for post-critically finite rational maps and non-preperiodic base points. For polynomials, this establishes a new class of known cases of a conjecture of…

We introduce the universal unitarily graded A-algebra for a commutative ring A and an arbitrary abelian extension U of the group of units of A, and use this concept to give simplified proofs of the main theorems of co-Galois theory in the…

For various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$.…

We provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(7)$. To my knowledge, this is the first instance of infinitely many such realizations for a perfect group which is…

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper provides an existence theorem for such fields over global fields with enough roots…

We present an astonishingly simple and elegant proof of the celebrated Basel problem.

In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois…

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This…

In this work we introduce new folding axioms involving easy 3D manoeuvres with the aim to push forward the arithmetic limits of the Huzita-Justin axioms. Those 3D axioms involve the use of a flat surface and the rigidity property of convex…

Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$,…

We show that finite Galois extensions with cyclic Galois group are radical.

In this paper we use the Galois module structure for the classical parameterizing spaces for elementary p-abelian extensions of a field K to give necessary and sufficient conditions for the solvability of any embedding problem which is an…

Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types…

A proof of the main theorem of the Galois theory is presented using the main theorem of symmetric polynomials. The idea originated from studying the "M\'emoire sur les conditions de r\'esolubilit\'e des \'equations par radicaux" of Evariste…

Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens H\"{o}chsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or…

Let $K/F$ be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified $p$-adic representation $\rho$ of the absolute Galois group of $K$ is determined…

We prove that certain fields have the property that their absolute Galois groups are free as profinite groups: the function field of a real curve with no real points; the maximal abelian extension of a 2-variable Laurent series field over a…

Zariski proved the general complex projective curve of genus g>6 is not rationally uniformized by radicals, that is, admits no map to the projective line whose Galois group is solvable. We give an example of a genus 7 complex projective…