Let [Formula: see text] denote the ring of algebraic integers of an algebraic number field [Formula: see text], where [Formula: see text] is a root of an irreducible trinomial [Formula: see text] belonging to [Formula: see text]. In this paper, we give necessary and sufficient conditions involving only [Formula: see text] for a given prime [Formula: see text] to divide the index of the subgroup [Formula: see text] in [Formula: see text]. In particular, we deduce necessary and sufficient conditions for [Formula: see text] to be equal to [Formula: see text].

Let q be a prime number and K = Q(θ) be an algebraic number field with θ a root of an irreducible trinomial x 6 +ax+b having integer coefficients. In this paper, we provide some explicit conditions on a, b for which K is not monogenic. As an application, in a special case when a = 0, K is not monogenic if b ≡ 7 mod 8 or b ≡ 8 mod 9. As an example, we also give a non-monogenic class of number fields defined by irreducible sextic trinomials.Throughout the paper, Z K denotes the ring of algebraic integers of an algebraic number field K. For a prime number q and a non-zero m belonging to the ring Z q of q-adic integers, v q (m) will be defined to be the highest power of q dividing m.

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In this article, we show that if ffalse(xfalse)=anxn+an−1xn−1+⋯+a0,a0≠0 is a polynomial with rational coefficients and if there exists a prime p whose highest power ri dividing ai (where ri=∞ if ai=0) satisfies rn=0, nri⩾false(n−ifalse)r0>0 for 0⩽i⩽n−1, then f(x) has at most gcd(r0,n) irreducible factors over the field boldQ of rational numbers and each irreducible factor has degree at least n/trueprefixgcdfalse(r0,nfalse) over boldQ. This result extends the famous Eisenstein–Dumas irreducibility criterion. In fact, we prove our result in a more general setup for polynomials with coefficients in arbitrary valued fields.