A124827 Order of Galois groups of irreducible Chebyshev polynomials of order n.

1, 2, 6, 8, 20, 12, 42, 16, 54, 40, 110, 48, 156, 84, 120, 64, 272, 108, 342, 160, 252

Offset: 1

Author

Artur Jasinski, Nov 09 2006

Keywords

• nonn
• uned
• more

Comments

All groups belonging to solvable Galois groups.

Very similar sequence is A002618 (disagreement occurred only for Chebyshev polynomials orders 8 and 16).

When the order of an irreducible Chebyshev polynomial is a prime number p, the Galois group is the Frobenius group of order p*(p-1) A036689.

In Magma classification the Galois groups are the following: T1_1, T2_1, T3_2, T4_3, T5_3, T6_3, T7_4, T8_8, T9_10, T11_4, T12_28, T13_6, T14_7, T15_11, T16_144, T17_5, T18_45, T19_6, T20_42, T21_15.

Is a(n) the order of Galois group of the polynomial x^n - 2? If so, then a(n) = n*phi(n) for n not divisible by 8, and n*phi(n)/2 otherwise (see the Math Overflow link below). Under this assumption, a(n) is multiplicative with a(p^e) = p^(2*e-1)*(p-1) for p being an odd prime; a(2) = 2, a(4) = 8, and a(2^e) = 2^(2*e-2) for e >= 3. - Jianing Song, Nov 22 2022

Examples

			a(5)=20 because the order of the Galois group of polynomial 16x^5-20x^3+5x-c is 20 (where c is an integer chosen so that the polynomial is irreducible). This transitive group is the Frobenius group F5 of order 20 (also called the metacyclic group M_5) T5_3(20) in Magma classification.
		

Links

• Math Overflow, Galois Group of x^n - 2

Crossrefs

Cf. A001710, A000142, A036689, A002618, A036689.

Cf. A127835.

Programs

Magma

Zx:=PolynomialRing(Integers()); f:=16*x^5-20*x^3+5*x-7; G:=GaloisGroup(f:Old); "Order of group",#G; // Juergen Klueners klueners(AT)math.uni-duesseldorf.de