Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.

Noether’s problem for GL(2, 3)

Abstract: We prove that Noether’s problem has an affirmative answer for the group GL(2, 3), over every field K. In particular, the group $$\widetilde{S_4}\cong {\rm GL}(2,3)$$ admits a generic polynomial over $$\mathbb Q$$ . As a consequence, so does the group $$\widetilde{S_5}$$ .

The inverse Galois problem over formal power series fields

Abstract: Consider a valuation ringR of a discrete Henselian field and a positive integerr. LetF be the quotient field of the ringR[[X 1, …,X r ]]. We prove that every finite group occurs as a Galois group overF. In particular, ifK 0 is an arbitrary field andr≥2, then every finite group occurs as a Galois group overK 0((X 1, …,X r )).

Number Fields with Discriminant ±2 a 3 b and Galois Group A n or S n

Abstract: The authors present three-point and four-point covers having bad reduction at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 12, 18, 28, and 33. By specializing these covers, they obtain number fields ramified at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 11, 12, 17, 18, 25, 28, 30, and 33.

On Tate-Shafarevich groups of abelian varieties

Abstract: Let K/F be a finite Galois extension of number fields with Galois group G, let A be an abelian variety defined over F, and let EU(A/K) and III(A/F) denote, respectively, the Tate-Shafarevich groups of A over K and of A over F. Assuming that these groups are finite, we derive, under certain restrictions on A and K/F, a formula for the order of the subgroup of IU(A/K) of G-invariant elements. As a corollary, we obtain a simple formula relating the orders of IU(A/K), IU(A/F) and 1U(A'F) when K/F is a quadratic extension and AX is the twist of A by the non-trivial character X of G.