A126270 a(n) = order of Galois group of the polynomial P(x) + n if P(x) + n (after dividing by the gcd of its coefficients) is irreducible, otherwise a(n) = 0, where P(x) = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 2.

8, 0, 0, 32, 32, 32, 16, 16, 32, 32, 32, 32, 32, 32, 16, 32, 16, 32, 32, 32, 32, 32, 32, 16, 32, 32, 32, 32, 32, 32, 16, 32, 32, 32, 0, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 8, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 16, 32, 32, 32, 32, 32, 32, 32, 16

Offset

0,1

Comments

P = 2*T_8(x/2), where T_8(x) is the degree 8 Chebyshev polynomial of the first kind.

For zeros in this sequence see A136362.

Links

Table of n, a(n) for n=0..70.

Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind

Example

Galois group of P+6 = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 8 is group 6 of degree 8 in MAGMA Transitive Group Identification, permutation group acting on a set of cardinality 8 with two generators, isomorphic to dihedral group D(8); it has index 2520 in symmetric group Sym(8) and order 16. Hence a(6) = 16.
a(34) = 0 since P+34 = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 36 = (x^4 - 8*x^2 + 18)*(x^4 + 2) is not irreducible.

Prog

(Magma) Zx<x>:=PolynomialRing(Integers()); T:=Coefficients(ChebyshevT(8)); P:=Zx ! [ 2^(2-i)*T[i]: i in [1..#T] ]; [ IsIrreducible(f) select Order(GaloisGroup(f)) else 0 where f is P+n: n in [0..70] ]; /* Klaus Brockhaus, Dec 27 2007 */
(Magma) Q:=RationalField(); R<x>:=PolynomialRing(Q); f:=x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 1; for n in {0 .. 30} do f:=f+1; if IsIrreducible(f) then Order(GaloisGroup(f)); else 0; end if; end for; /* N. J. A. Sloane, Dec 28 2007 */

Crossrefs

Cf. A124827, A126271, A136362.

Sequence in context: A341807 A340915 A028701 * A371665 A169696 A192059

Adjacent sequences: A126267 A126268 A126269 * A126271 A126272 A126273

Keyword

nonn

Author

Artur Jasinski, Dec 23 2006

Extensions

Edited and extended by Klaus Brockhaus, Dec 27 2007

Status

approved