Extending this list, we find η(5i)=η(i)√5(1+√52)−1/2η(6i)=η(i)63/8(5−√32−33/4√2)1/6η(7i)=η(i)√7(−72+√7+12√−7+4√7)1/4η(8i)=η(i)241/32(−1+4√2)1/2(1+√2)1/8η(16i)=η(i)2113/64(−1+4√2)1/4(1+√2)1/16(−25/8+√1+√2)1/2
It seems that for prime p=4m+1, then x=(√pη(pi)η(i))2 is an algebraic number of degree p−12, while if p=4m+3 for p>3, then y=(√pη(pi)η(i))4 has degree p+12. For the special case p=7, note that y=(−72+√7+12√−7+4√7)≈0.092192
is one of the four quartic roots such that the polynomial on the RHS vanishes (y2+5y+1)3(y2+13y+49)y=123+(y4+14y3+63y2+70y−7)2y
and the LHS assumes the value 123=1728.
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