OFFSET
1,1
COMMENTS
Condition on odd prime p so that Q(Cp) is not rational over Q: p = 8q+1 where q != -1 (mod 4), q is squarefree, and any of p-q and p-4q is not square.
LINKS
Shizuo Endo and Takehiko Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan, Volume 25, Number 1 (1973), 1-167 (see Proposition 3.6 (ii) p.18).
Akinari Hoshi, On Noether's problem for cyclic groups of prime order, arXiv:1402.3678 [math.NT], 2014 (see Proposition 3.1 (ii) p.4 and Table 2 p.19).
MATHEMATICA
Reap[For[p = 3, p < 5000, p = NextPrime[p], If[Mod[p, 8] == 1 && Mod[q = Quotient[p, 8], 4] != 3 && SquareFreeQ[q] && AllTrue[{p-q, p-4q}, !IntegerQ[Sqrt[#]]&], Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 15 2018 *)
PROG
(PARI) isok(p) = isprime(p) && ((type(q = (p-1)/8)== "t_INT") && ((q % 4) != 3) && (!issquare(p-q)) && (!issquare(p-4*q)));
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Apr 08 2014
STATUS
approved