A344407 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

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

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).

EXAMPLE

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.

PROG

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

CROSSREFS

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

Sequence in context: A356855 A335913 A340667 * A197243 A175372 A069191

Adjacent sequences: A344404 A344405 A344406 * A344408 A344409 A344410

KEYWORD

sign,easy

AUTHOR

Jianing Song, May 17 2021

STATUS

approved