A344407 Discriminant of the (2n)-th cyclotomic field Q(zeta_(2n)). Equivalently, discriminant of the (2n)-th cyclotomic polynomial.

1, -4, -3, 256, 125, 144, -16807, 16777216, -19683, 4000000, -2357947691, 5308416, 1792160394037, 1157018619904, 1265625, 18446744073709551616, 2862423051509815793, 1586874322944, -5480386857784802185939, 1048576000000000000, 205924456521, 5829995856912430117421056

Offset: 1

Author

Jianing Song, May 17 2021

Keywords

• sign
• easy

Comments

Note that Q(zeta_n) = Q(zeta_(2n)) for odd n, so this sequence is A004124 with redundant values removed.

a(n) is negative <=> phi(2n) == 2 (mod 4) <=> n = 2 or n is of the form p^e, where p is a prime congruent to 3 modulo 4.

Examples

			n = 2: Q(zeta_4) = Q(i) has discriminant -4;
n = 3: Q(zeta_6) = Q(sqrt(-3)) has discriminant -3;
n = 4: Q(zeta_8) = Q(sqrt(2), i) has discriminant 256;
n = 5: Q(zeta_10) = Q(exp(2*Pi*i/5)) has discriminant 125;
n = 6: Q(zeta_12) = Q(sqrt(3), i) has discriminant 144.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A004124, A062570 (degree of Q(zeta_(2n))).

Programs

PARI

vector(25,n,poldisc(polcyclo(2*n)))

Formula

a(n) = A004124(2n). See A004124 for its formula.

For n >= 2, a(p) = (-1)^((p-1)/2) * A130614(n), where p = prime(n).