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1. CMB Online first
Infinite Families of $A_4$-Sextic Polynomials In this article we develop a test to determine whether a sextic
polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic
to the alternating group $A_4$. This test does not involve the
computation of resolvents, and we use this test to construct several
infinite families of such polynomials.
Keywords:Galois group, sextic polynomial, inverse Galois theory, irreducible polynomial Categories:12F10, 12F12, 11R32, 11R09 |
2. CMB 2011 (vol 56 pp. 510)
Linear Forms in Monic Integer Polynomials We prove a necessary and sufficient condition on the list of
nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic
polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form
$u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in
\mathbb{Z}[x]$. This condition is independent of $f$. We also show that if
this condition holds, then the monic polynomials $f_1,\dots,f_k$
can be chosen to be irreducible in $\mathbb{Z}[x]$.
Keywords:irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion Categories:11R09, 11C08, 11B83 |
3. CMB 2009 (vol 52 pp. 511)
The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value We use some classical estimates for polynomial roots to provide
several irreducibility criteria for polynomials with integer
coefficients that have one sufficiently large coefficient and take a
prime value.
Keywords:Estimates for polynomial roots, irreducible polynomials Categories:11C08, 11R09 |