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We show that a nonempty family of n-generated subgroups of a pro-p group has a maximal element. This suggests that 'Noetherian Induction' can be used to discover new features of finitely generated subgroups of pro-p groups. To demonstrate this, we show that in various pro-p groups Γ (e.g. free pro-p groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup H = 1 is the greatest subgroup of Γ containing H as an open subgroup. We also show that an ascending sequence of n-generated subgroups of a limit group must terminate (this extends the analogous result for free groups proved by Takahasi, Higman, and Kapovich-Myasnikov).

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of N > deg c + deg d such that the polynomialare polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that x N − kx 2 + 1 is irreducible for all N ≥ 5 and k ∈ {3, 4, . . . , 24} \ {9, 16}. We also give a complete description of the factorization of polynomials of the form x N + kx N−1 ± (lx + 1) with k, l ∈ Z, k = l.

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