Let ${\displaystyle K}$ be a number field of degree ${\displaystyle n}$, ${\displaystyle {\mathbb{Z}}_K}$ its ring of integers, and ${\displaystyle d_K}$ its absolute discriminant. It is well known that ${\displaystyle {\mathbb{Z}}_K}$ is a free abelian group of rank ${\displaystyle n}$ and by the fundamental theorem of finite abelian groups, ${\displaystyle ({\mathbb{Z}}_K:\mathbb{Z}\left[\theta \right])}$ is a finite group for every primitive element ${\displaystyle \theta \in {\mathbb{Z}}_K}$ of ${\displaystyle K}$. Let ${\displaystyle ind\left(\theta \right)=({\mathbb{Z}}_K:\mathbb{Z}\left[\theta \right])}$. ${\displaystyle ind\left(\theta \right)}$ is called the index of ${\displaystyle \theta }$. The index of the number field ${\displaystyle K}$ is defined by ${\displaystyle i\left(K\right)=GCD(({\mathbb{Z}}_K:\mathbb{Z}\left[\theta \right])\mid K=\mathbb{Q}\left(\theta \right)and\theta \in {\mathbb{Z}}_K)}$. A rational prime integer ${\displaystyle p}$ dividing ${\displaystyle i\left(K\right)}$ is called a prime common index divisor of ${\displaystyle K}$. The number field ${\displaystyle K}$ is called monogenic if it admits a ${\displaystyle \mathbb{Z}}$ basis of type ${\displaystyle (1,\theta ,\dots ,{\theta }^{n-1})}$ for some ${\displaystyle \theta \in {\mathbb{Z}}_K}$. Remark that if ${\displaystyle {\mathbb{Z}}_K}$ has a power integral basis, then ${\displaystyle i\left(K\right)=1}$. Therefore a field having a prime common index divisor is not monogenic. Monogenity of number fields is a classical problem of algebraic number theory, going back to Dedekind, Hasse and Hensel, see for instance [19,25,26] for the present state of this area. It is called a problem of Hasse to give an arithmetic characterization of those number fields which are monogenic [23,25,26]. For any primitive element ${\displaystyle \theta \in {\mathbb{Z}}_K}$ of ${\displaystyle K}$, it is well-known that where ${\displaystyle ▵\left(\theta \right)}$ is the discriminant of the minimal polynomial of ${\displaystyle \theta }$ over ${\displaystyle \mathbb{Q}}$ [19].

Clearly, ${\displaystyle ind\left(\theta \right)=1}$ for some primitive element ${\displaystyle \theta \in {\mathbb{Z}}_K}$ of ${\displaystyle K}$ if and only if ${\displaystyle (1,\theta ,\dots ,{\theta }^{n-1})}$ is a power integral basis of ${\displaystyle {\mathbb{Z}}_K}$.

The problem of testing the monogenity of number fields and constructing power integral bases have been intensively studied during the last four decades mainly by Gaál, Györy, Nakahara, Pohst and their collaborators (see for instance [1,16,32]). In 1871, Dedekind was the firstone who gave an example of a number field with non trivial index, he considered the cubic field ${\displaystyle K}$ generated by a root of ${\displaystyle x^3-x^2-2x-8}$ and showed that the rational prime ${\displaystyle 2}$ splits completely in ${\displaystyle K}$ ([5, § 5, page 30]). According to a well known theorem of Dedekind ([24, Chapter I, Proposition 8.3]), if we suppose that ${\displaystyle K}$ is monogenic, then we would be able to find a cubic polynomial defining ${\displaystyle K}$, that splits completely into distinct polynomials of degree ${\displaystyle 1}$ in ${\displaystyle {\mathbb{F}}_2\left[x\right]}$. Since there is only two distinct polynomials of degree ${\displaystyle 1}$ in ${\displaystyle {\mathbb{F}}_2\left[x\right]}$, this is impossible. In ${\displaystyle 1930}$, Engstrom was the first one who related the prime ideal factorization and the index of a number field of degree less than ${\displaystyle 8}$ [13]. For any number field ${\displaystyle K}$ of degree ${\displaystyle n\le 7}$, he showed that ${\displaystyle {\nu }_p\left(i\left(K\right)\right)}$ is explicitly determined by the factorization of ${\displaystyle p{\mathbb{Z}}_K}$ into powers of prime ideals of ${\displaystyle K}$ for every positive rational prime integer ${\displaystyle p\le n}$. This motivated Narkiewicz to ask a very important question, stated as problem 22 in Narkiewicz’s book ([31, Problem 22]), which asks for an explicit formula of the highest power ${\displaystyle {\nu }_p\left(i\left(K\right)\right)}$ for a given rational prime ${\displaystyle p}$ dividing ${\displaystyle i\left(K\right)}$. In [30], Nakahara studied the index of non-cyclic but abelian biquadratic number fields. He showed that the field index of such fields is in the set ${\displaystyle \{1,2,3,4,6,12\}}$. In [17] Gaál et al. characterized the field indices of biquadratic number fields having Galois group ${\displaystyle V_4}$ and they proved that ${\displaystyle i\left(K\right)\in \{1,2,3,4,6,12\}}$. Recently, many authors are interested on monogenity of number fields defined by trinomials. Davis and Spearman [6] studied the index of quartic number fields ${\displaystyle K}$ generated by a root of such a quartic trinomial ${\displaystyle F\left(x\right)=x^4+ax+b\in \mathbb{Z}\left[x\right]}$. They gave necessary and sufficient conditions on ${\displaystyle a}$ and ${\displaystyle b}$ so that a prime ${\displaystyle p}$ is a common index divisor of ${\displaystyle K}$ for ${\displaystyle p=2,3}$. Their method is based on the calculation of the ${\displaystyle p}$-index form of ${\displaystyle K}$, using ${\displaystyle p}$-integral bases of ${\displaystyle K}$. El Fadil and Gaál [11] studied the index of quartic number fields ${\displaystyle K}$ generated by a root of a quadratic trinomial of the form ${\displaystyle F\left(x\right)=x^4+a{x}^2+b\in \mathbb{Z}\left[x\right]}$. They gave necessary and sufficient conditions on ${\displaystyle a}$ and ${\displaystyle b}$ so that a prime ${\displaystyle p}$ is a common index divisor of ${\displaystyle K}$ for every prime integer ${\displaystyle p}$. In [15], for a sextic number field ${\displaystyle K}$ defined by a trinomial ${\displaystyle F\left(x\right)=x^6+a{x}^3+b\in \mathbb{Z}\left[x\right]}$, Gaál studied the multi-monegenity of ${\displaystyle K}$; he calculated all possible power integral bases of ${\displaystyle K}$. In [9], we extended Gaál’s studies by providing some cases where ${\displaystyle K}$ is not monogenic. Also in [10], for every prime integer ${\displaystyle p}$, we gave necessary and sufficient conditions on ${\displaystyle a}$ and ${\displaystyle b}$ so that ${\displaystyle p}$ is a common index divisor of ${\displaystyle K}$, where ${\displaystyle K}$ is a number field defined by an irreducible trinomial ${\displaystyle F\left(x\right)=x^5+a{x}^2+b\in \mathbb{Z}\left[x\right]}$. In [8], we provided some sufficient conditions which guarantee that ${\displaystyle i\left(K\right)}$ is not trivial, and so ${\displaystyle K}$ is not mongenic. In this paper, for a septic number field generated by a root of a trinomial ${\displaystyle F\left(x\right)=x^7+ax+b\in \mathbb{Z}\left[x\right]}$ and for every prime integer ${\displaystyle p}$, we calculate ${\displaystyle {\nu }_p\left(i\left(K\right)\right)}$, the highest power of ${\displaystyle p}$ dividing the index ${\displaystyle i\left(K\right)}$ of the field ${\displaystyle K}$. Our method is based on Newton’s polygon techniques applied in prime ideal factorization, which is performed in [21,22] and in Montes’ thesis defended in 1999. The author is very thankful to Professor Enric Nart who provided him a copy of Montes’ thesis.

Throughout this section ${\displaystyle K}$ is a number field generated by a root ${\displaystyle \alpha }$ of an irreducible trinomial ${\displaystyle F\left(x\right)=x^7+ax+b\in \mathbb{Z}\left[x\right]}$ and we assume that for every rational prime integer ${\displaystyle p}$, ${\displaystyle {\nu }_p\left(a\right)\le 5}$ or ${\displaystyle {\nu }_p\left(b\right)\le 6}$. Along this paper, for every integer ${\displaystyle a\in \mathbb{Z}}$ and a prime integer ${\displaystyle p}$, let ${\displaystyle a_p=\frac{a}{p^{{\nu }_p\left(a\right)}}}$.

We start with the following theorem, which characterizes when is ${\displaystyle \mathbb{Z}\left[\alpha \right]}$ integrally closed?

The following example gives an infinite family of monogenic septic number fields defined by non monogenic trinomials.

In the remainder of this section, for every prime integer ${\displaystyle p}$ and for every values of ${\displaystyle a}$ and ${\displaystyle b}$, we calculate ${\displaystyle {\nu }_p\left(i\left(K\right)\right)}$. For every integers ${\displaystyle a}$ and ${\displaystyle b}$, let ${\displaystyle ▵=-(6^6{a}^7+7^7{b}^6)}$ be the discriminant of ${\displaystyle F\left(x\right)}$ and for every prime integer ${\displaystyle p}$, let ${\displaystyle {▵}_p=\frac{▵}{p^{{\nu }_p(▵)}}}$.

In 1894, Hensel developed a powerful approach by showing that for every prime integer ${\displaystyle p}$, the prime ideals of ${\displaystyle {\mathbb{Z}}_K}$ lying above ${\displaystyle p}$ are in one–one correspondence with monic irreducible factors of ${\displaystyle F\left(x\right)}$ in ${\displaystyle {\mathbb{Q}}_p\left[x\right]}$. For every prime ideal corresponding to any irreducible factor in ${\displaystyle {\mathbb{Q}}_p\left[x\right]}$, the ramification index and the residue degree together are the same as those of the local field defined by the associated irreducible factor [28]. Since then, to factorize ${\displaystyle p{\mathbb{Z}}_K}$, we need to factorize ${\displaystyle F\left(x\right)}$ in ${\displaystyle {\mathbb{Q}}_p\left[x\right]}$. Newton’s polygon techniques can be used to refine the factorization. This is a standard method which is rather technical but very efficient to apply. We have introduced the corresponding concepts in several former papers. Here we only give a brief introduction which makes our proofs understandable. For a detailed description, we refer to Ore’s Paper [34] and Guardia, Montes and Nart’s paper [20]. For every prime integer ${\displaystyle p}$, let ${\displaystyle {\nu }_p}$ be the ${\displaystyle p}$-adic valuation of ${\displaystyle {\mathbb{Q}}_p}$ and ${\displaystyle {\mathbb{Z}}_p}$ the ring of ${\displaystyle p}$-adic integers. Let ${\displaystyle F\left(x\right)\in {\mathbb{Z}}_p\left[x\right]}$ be a monic polynomial and ${\displaystyle \varphi \in {\mathbb{Z}}_p\left[x\right]}$ a monic lift of an irreducible factor of ${\displaystyle \overline{F\left(x\right)}}$ modulo ${\displaystyle p}$. Let ${\displaystyle F\left(x\right)=a_0\left(x\right)+a_1\left(x\right)\varphi \left(x\right)+\cdots +a_n\left(x\right)\varphi {\left(x\right)}^l}$ be the ${\displaystyle \varphi }$-expansion of ${\displaystyle F\left(x\right)}$, ${\displaystyle N_{\varphi }\left(F\right)}$ the ${\displaystyle \varphi }$-Newton polygon of ${\displaystyle F\left(x\right)}$ and ${\displaystyle N_{\varphi }^{+}\left(F\right)}$ its principal part. Let ${\displaystyle {\mathbb{F}}_{\varphi }}$ be the field ${\displaystyle {\mathbb{F}}_p\left[x\right]/\left(\overline{\varphi }\right)}$. For every side ${\displaystyle S}$ of ${\displaystyle N_{\varphi }^{+}\left(F\right)}$ with length ${\displaystyle l}$ and initial point ${\displaystyle (s,u_s)}$, for every ${\displaystyle i=0,\dots ,l}$, let ${\displaystyle c_i\in {\mathbb{F}}_{\varphi }}$ be the residue coefficient, defined as follows: Let ${\displaystyle -\lambda =-h/e}$ be the slope of ${\displaystyle S}$, where ${\displaystyle h}$ and ${\displaystyle e}$ are two positive coprime integers. Then ${\displaystyle d=l/e}$ is the degree of ${\displaystyle S}$. Let ${\displaystyle R_1\left(F\right)\left(y\right)=t_d{y}^d+t_{d-1}{y}^{d-1}+\cdots +t_{1}y+t_0\in {\mathbb{F}}_{\varphi }\left[y\right]}$, called the residual polynomial of ${\displaystyle F\left(x\right)}$ associated to the side ${\displaystyle S}$, where for every ${\displaystyle i=0,\cdots ,d}$, ${\displaystyle t_i=c_{ie}}$. If ${\displaystyle R_1\left(F\right)\left(y\right)}$ is square free for each side of the polygon ${\displaystyle N_{\varphi }^{+}\left(F\right)}$, then we say that ${\displaystyle F\left(x\right)}$ is ${\displaystyle \varphi }$-regular.

Let ${\displaystyle \overline{F\left(x\right)}=\prod _{i=1}^r{\overline{{\varphi }_i}}^{l_i}}$ be the factorization of ${\displaystyle F\left(x\right)}$ into powers of monic irreducible coprime polynomials over ${\displaystyle {\mathbb{F}}_p}$, we say that the polynomial ${\displaystyle F\left(x\right)}$ is ${\displaystyle p}$-regular if ${\displaystyle F\left(x\right)}$ is a ${\displaystyle {\varphi }_i}$-regular polynomial with respect to ${\displaystyle p}$ for every ${\displaystyle i=1,\cdots ,r}$. Let ${\displaystyle N_{{\varphi }_i}^{+}\left(F\right)=S_{i1}+\cdots +S_{i{r}_i}}$ be the ${\displaystyle {\varphi }_i}$-principal Newton polygon of ${\displaystyle F\left(x\right)}$ with respect to ${\displaystyle p}$. For every ${\displaystyle j=1,\cdots ,r_i}$, let ${\displaystyle R_{1_{ij}}\left(F\right)\left(y\right)=\prod _{s=1}^{s_{ij}}{\psi }_{ijs}^{a_{ijs}}\left(y\right)}$ be the factorization of ${\displaystyle R_{1_{ij}}\left(F\right)\left(y\right)}$ in ${\displaystyle {\mathbb{F}}_{{\varphi }_i}\left[y\right]}$, where ${\displaystyle R_{1_{ij}}\left(F\right)\left(y\right)}$ is the residual polynomial of ${\displaystyle F\left(x\right)}$ attached to the side ${\displaystyle S_{ij}}$. Then we have the following theorem of index of Ore:

The Dedekind criterion can be reformulated as follows:

When the theorem of Ore fails, that is ${\displaystyle F\left(x\right)}$ is not ${\displaystyle p}$-regular, then in order to complete the factorization of ${\displaystyle F\left(x\right)}$, Guardia, Montes, and Nart introduced the notion of high order Newton polygon. By analogous to the first order, for each order ${\displaystyle r}$, the authors of [20] introduced the valuation ${\displaystyle {\omega }_2}$ of order ${\displaystyle r}$, the key polynomial ${\displaystyle {\varphi }_2\left(x\right)}$ of such a valuation, ${\displaystyle N_r\left(F\right)}$ the Newton polygon of any polynomial ${\displaystyle F\left(x\right)}$ with respect to ${\displaystyle {\omega }_2}$ and ${\displaystyle {\varphi }_2\left(x\right)}$, and for every side of ${\displaystyle N_r\left(F\right)}$ the residual polynomial ${\displaystyle R_r\left(F\right)}$, and the index of ${\displaystyle F\left(x\right)}$ in order ${\displaystyle r}$. For more details, we refer to [20].

For the proofs of Theorems 2.3 and 2.4, we need the following lemma, which characterizes the prime common index divisors of ${\displaystyle K}$.

By virtue of Engstrom’s results [14], the proof is done if we provide the factorization of ${\displaystyle 2{\mathbb{Z}}_K}$ into powers of prime ideals of ${\displaystyle {\mathbb{Z}}_K}$. Based on Theorem 2.1, we deal with the cases: ${\displaystyle 2|a}$ and ${\displaystyle 4|b}$ or ${\displaystyle (a,b)\in \left\{\right(1,2),(3,0\left)\right\}(\mathrm{mod}4)}$.