Octahedral extensions with a given cubic subfield

Arithmetic of quaternion origami

“…Then, product 2 = v 3 v 4 = e 8 1 e 4 4 −2e 7 3 1 5 2 a 9 +2 9 3 5 5 1 a 8 z 2 +2 9 3 6 5 1 a 7 bz +2 12 3 4 17 1 a 7 z 4 −2 13 3 1 5 1 a 7 zw 2 +2 5 3 5 5 1 41 1 a 6 b 2 + 2 13 3 5 17 1 a 6 bz 3 − 2 12 3 2 5 1 a 6 bw 2 + 2 12 3 6 a 6 z 6 − 2 15 3 4 a 6 z 3 w 2 + 2 8 3 7 151 1 a 5 b 2 z 2 + 2 12 3 8 a 5 bz 5 − 2 14 3 6 a 5 bz 2 w 2 + 2 8 3 7 181 1 a 4 b 3 z+2 11 3 7 31 1 a 4 b 2 z 4 −2 12 10 3 2 5 2 a 10 z − 2 9 3 3 5 2 a 9 b − 2 11 3 3 5 1 19 1 a 9 z 3 − 2 10 3 5 5 1 19 1 a 8 bz 2 − 2 16 3 5 a 8 z 5 +2 16 3 2 5 1 a 8 z 2 w 2 −2 8 3 6 5 1 43 1 a 7 b 2 z −2 15 3 6 5 1 a 7 bz 4 +2 16 3 3 5 1 a 7 bzw 2 +2 17 3 4 a 7 z 4 w 2 −2 7 3 6 5 1 53 1 a 6 b 3 − 2 10 3 6 5 2 23 1 a 6 b 2 z 3 + 2 14 3 4 5 1 a 6 b 2 w 2 + 2 18 3 5 a 6 bz 3 w 2 − 2 9 3 9 5 1 17 1 a 5 b 3 z 2 − 2 14 3 8 a 5 b 2 z 5 + 2 15 3 5 23 1 a 5 b 2 z 2 w 2 − 2 6 3 9 5 2 29 1 a 4 b 4 z − 2 13 3 9 5 1 a 4 b 3 z 4 + 2 15 3 6 11 1 a 4 b 3 zw 2 + 2 15 3 7 a 4 b 2 z 4 w 2 − 2 5 3 9 647 1 a 3 b 5 − 2 7 3 9 5 1 211 1 a 3 b 4 z 3 + 2 16 3 7 a 3 b 4 w 2 + 2 16 3 8 a 3 b 3 z 3 w 2 + 2 2 5 1 a 3 (v 1 + v 2 ) − 2 6 3 11 5 1 83 1 a 2 b 5 z 2 + 2 12 3 8 41 1 a 2 b 4 z 2 w 2 + 2 4 3 3 a 2 z 2 (v 1 + v 2 ) − 2 4 3 11 5 1 311 1 ab 6 z + 2 12 3 9 17 1 ab 5 zw 2 + 2 4 3 4 abz(v 1 + v 2 ) − 2 2 3 2 az(v 3 + v 4 ) − 2 3 3 12 11 1 37 1 b 7 + 2 10 3 10 11 1 b 6 w 2 + 3 3 41 1 b 2 (v 1 +v 2 )−2 1 3 3 b(v 3 +v 4 )x 5 +(2 8 3 2 5 2 a 12 +2 14 3 3 5 1 a 11 z 2 +2 14 3 4 5 1 a 10 bz +2 18 3 4 a 10 z 4 −2 15 3 2 5 1 a 10 zw 2 + 2 8 3 6 5 1 7 1 a 9 b 2 + 2 19 3 5 4a 9 bz 3 − 2 14 3 3 5 1 a 9 bw 2 − 2 20 3 3 a 9 z 3 w 2 + 2 12 3 7 37 1 a 8 b 2 z 2 − 2 19 3 5 a 8 bz 2 w 2 + 2 20 3 2 a 8 z 2 w 4 + 2 12 3 7 47 1 a 7 b 3 z+2 17 3 7 a 7 b 2 z 4 −2 13 3 6 37 1 a 7 b 2 zw 2 +2 20 3 3 a 7 bzw 4 +2 5 3 8 11 1 97 1 a 6 b 4 +2 18 3 8 a 6 b 3 z 3 −2 12 15 3 11 a 2 b 5 z 2 w 2 +2 16 3 8 a 2 b 4 z 2 w 4 −2 5 3 5 a 2 bz 2 (v 1 +v 2 )+2 4 3 3 a 2 z 2 (v 3 +v 4 )+2 8 3 13 37 1 ab 7 z −2 9 3 11 101 1 ab 6 zw 2 + 2 16 3 9 ab 5 zw 4 −2 3 3 4 23 1 ab 2 z(v 1 +v 2 )+2 4 3 4 abz(v 3 +v 4 )+3 14 37 2 1b 8 −2 8 3 12 37 1 b 7 w 2 +2 14 3 10 b 6 w 4 −2 2 3 5 11…”

“…The main theorem of this article is that the splitting field of g over Q is also contained in the splitting field of f E,Q8,P over Q. Once there are two common S 4 -extensions of Q containing the 2-division field of E, there is also a third common S 4 -extension of Q [3, Proposition 1.1], [9,Theorem 2.4].…”