PSL(2,7) septimic fields with a power basis

A note on families of monogenic number fields

“…It should be understood that using Theorem 4, many more examples can be produced, using polynomials from the literature. Also, several existing results, such as those of [10], [11] or [16], can be regained as immediate corollaries of Theorem 4.…”

“…In particular, we show: Theorem 6. Let G be one of P GL 2 (7), AGL 3 (2) or P SL 2 (11). Then there are infinitely many monogenic number fields with Galois group (of the Galois closure) isomorphic to G.…”

“…4.2. The group P SL 2 (11). Let G be the projective special linear group P SL 2 (11), in its (exceptional) transitive permutation action on 11 points.…”

“…Proposition 8. The polynomial f (t, x) := x 11 − 3x 10 + 7x 9 − 25x 8 + 46x 7 − 36x 6 + 60x 4 − 121x 3 + 140x 2 − 95x + 27 + tx 2 (x − 1) 3 ∈ Z[t, x] has Galois group P SL 2 (11) over Q(t).…”

“…It is unclear for which finite groups there are infinitely many monogenic number fields with Galois group (of the Galois closure) isomorphic to G. While there are infinitely many examples for the symmetric group S n , it has been shown for cyclic groups G of prime order > 5 that there is at most one monogenic G-number field ( [8]). Several recent works have obtained infinite families of monogenic fields for prescribed Galois groups, including quintic D 5 -fields ( [10]), sextic A 4 -( [5]) and P SL 2 (5)fields ( [16]) and septimic P SL 2 (7)-fields ( [11]).…”