# Monogenity and Power Integral Bases: Recent Developments

## Abstract

**:**

## 1. Introduction

## 2. Tools

#### 2.1. Dedekind’s Criterion

**Theorem**

**1**

**.**

**Theorem**

**2**

- 1.
- p does not divide the index $I\left(\alpha \right)=({\mathbb{Z}}_{K}:\mathbb{Z}\left[\alpha \right])$.
- 2.
- For every $i=1,\cdots ,r$, either ${\mathcal{l}}_{i}=1$ or ${\mathcal{l}}_{i}\ge 2$ and $\overline{{\varphi}_{i}\left(x\right)}$ does not divide $\overline{M\left(x\right)}$ in ${\mathbb{F}}_{p}\left[x\right]$.

#### 2.2. The Field Index

**Theorem**

**3**

**.**

- The prime factors of the field index are smaller than the degree of the field.

**Theorem**

**4**

**.**

#### 2.3. Newton Polygon Method

**Theorem**

**5**

**.**

- 1.
- We have$${\nu}_{p}\left(\mathrm{ind}\left(f\right)\right)\ge \sum _{i=1}^{r}{\mathrm{ind}}_{{\varphi}_{i}}\left(f\right).$$The equality holds if $f\left(x\right)$ is p-regular.
- 2.
- If $f\left(x\right)$ is p-regular, then$$p{\mathbb{Z}}_{K}=\prod _{i=1}^{r}\prod _{j=1}^{{r}_{i}}\prod _{k=1}^{{s}_{ij}}{\mathfrak{p}}_{ijk}^{{e}_{ij}},$$is the factorization of $p{\mathbb{Z}}_{K}$ into powers of prime ideals of ${\mathbb{Z}}_{K}$ lying above p, where ${e}_{ij}={\mathcal{l}}_{ij}/{d}_{ij}$, ${\mathcal{l}}_{ij}$ is the length of ${S}_{ij}$, ${d}_{ij}$ is the ramification degree of ${S}_{ij}$, and ${f}_{ijk}=deg\left({\varphi}_{i}\right)\times deg\left({\psi}_{ijk}\right)$ is the residue degree of the prime ideal ${\mathfrak{p}}_{ijk}$ over p.

#### 2.4. Algorithmic Methods

**Theorem**

**6**

**.**

## 3. Results

#### 3.1. Pure Fields, Trinomials, Quadrinomials, etc.

- Z. S. Aygin and K. D. Nguyen [24]: $n=3$;
- L. El Fadil [25]: $n=12$;
- L. El Fadil [26]: $n=18$;
- L. El Fadil [27]: $n=20$;
- L. El Fadil [28]: $n=24$;
- L. El Fadil [29]: $n=36$;
- Fadil, L.E. H. Ben Yakkou and J. Didi [30]: $n=42$;
- Fadil, L.E. H. Choulli and O. Kchit [31]: $n=60$;
- L. El Fadil and M. Faris [32]: $n=84$;
- H. Ben Yakkou and O. Kchit [33]: $n={3}^{k}$;
- L. El Fadil [34]: $n=2\xb7{3}^{k}$;
- L. El Fadil [35]: $n=6,{2}^{k}\xb7{3}^{\mathcal{l}}$;
- Yakkou, H.B. A. Chillali and L. El Fadil [36]: $n={2}^{k}\xb7{5}^{\mathcal{l}}$;
- L. El Fadil [37]: $n={3}^{k}\xb7{7}^{\mathcal{l}}$;
- L. El Fadil and A. Najim [38]: $n={2}^{k}\xb7{3}^{\mathcal{l}}$;
- L. El Fadil and O. Kchit [39]: $n={2}^{k}\xb7{7}^{\mathcal{l}}$;
- L. El Fadil [40]: $n={2}^{k}\xb7{3}^{\mathcal{l}}\xb7{5}^{t}$;
- H. Ben Yakkou and L. El Fadil [41]: $n={p}^{k}$;

**Theorem**

**7**

**.**

- L. El Fadil [45]: ${x}^{4}+ax+b$;
- L. El Fadil and I. Gaál [46]: ${x}^{4}+a{x}^{2}+b$;
- H. Smith [47]: ${x}^{4}+ax+b,{x}^{4}+c{x}^{3}+d$;
- L. Jones [48] showed that there exist exactly three distinct monogenic trinomials of the form ${x}^{4}+b{x}^{2}+d$ with Galois ${C}_{4}$;
- Jakhar, A. S. Kaur and S. Kumar [49]: ${x}^{5}+ax+b$;
- L. El Fadil [50]: ${x}^{5}+a{x}^{2}+b$;
- L. El Fadil [51]: ${x}^{5}+a{x}^{3}+b$;
- L. El Fadil [52]: ${x}^{6}+ax+b$;
- A. Jakhar and S. Kumar [53]: ${x}^{6}+ax+b$;
- L. El Fadil [54]: ${x}^{6}+a{x}^{3}+b$;
- L. El Fadil and O. Kchit [55]: ${x}^{6}+a{x}^{5}+b$;
- A. Jakhar and S. Kaur [56]: ${x}^{6}+a{x}^{m}+b$;
- R. Ibarra, H. Lembeck, M. Ozaslan, H. Smith and K. E. Stange [57]: ${x}^{n}+ax+b,{x}^{n}+c{x}^{n-1}+d$ for $n=5,6$;
- L. El Fadil and O. Kchit [58]: ${x}^{7}+a{x}^{3}+b$;
- H. Ben Yakkou [59]: ${x}^{7}+a{x}^{5}+b$;
- Jakhar, A. S. Kaur and S. Kumar [60]: ${x}^{7}+ax+b$;
- H. Ben Yakkou [61]: ${x}^{8}+ax+b$;
- H. Ben Yakkou and B. Boudine [62]: ${x}^{8}+ax+b$;
- Jakhar, A. S. Kaur and S. Kumar [63]: ${x}^{8}+a{x}^{m}+b$;
- L. Jones [64] considered monogenic trinomials of type ${x}^{8}+a{x}^{4}+b$ with prescribed Galois group;
- O. Kchit [65]: ${x}^{9}+ax+b$;
- H. Ben Yakkou and P. Tiebekabe [66]: ${x}^{9}+ax+b$;
- L. El Fadil and O. Kchit [67]: ${x}^{9}+a{x}^{2}+b$;
- L. El Fadil and O. Kchit [68]: ${x}^{12}+a{x}^{m}+b$;
- H. Ben Yakkou [69]: ${x}^{{2}^{r}}+a{x}^{m}+b$;
- H. Ben Yakkou and L. El Fadil [70]: ${x}^{n}+ax+b,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n=5,6,{3}^{k},{2}^{k}\xb7{3}^{\mathcal{l}},{2}^{k}\xb7{3}^{\mathcal{l}}+1$;
- A. Jakhar and S. Kumar [71] gave explicit conditions for the non-monogenity of ${x}^{{q}^{s}}-ax-b$;
- A. Jakhar [72]: ${x}^{{p}^{s}}-a{x}^{m}-b$;
- B. Jhorar and S. K. Khanduja [73]: ${x}^{n}+ax+b$, showed also that $f\left(x\right)={x}^{n}-x-1$ is monogenic, if and only if $\left|D\left(f\right)\right|={n}^{n}-{(n-1)}^{n-1}$ is squarefree;
- H. Ben Yakkou [74]: ${x}^{n}+a{x}^{m}+b,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n={p}^{k},s\xb7{p}^{k},{2}^{k}\xb7{3}^{\mathcal{l}}$;
- L. El Fadil [75]: ${x}^{n}+a{x}^{m}+b,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n={2}^{k}\xb7{3}^{\mathcal{l}}$;
- A. Jakhar [76]: ${x}^{n}-a{x}^{m}-b$;
- Jakhar, A. S. Khanduja and N. Sangwan [77]: ${x}^{n}+a{x}^{m}+b$;
- Jakhar, A. S. Khanduja and N. Sangwan [78] gave necessary and sufficient conditions in terms of $a,b,m,n$ for a given prime, p, to divide $I\left(\vartheta \right)$, where $\vartheta $ is a root of ${x}^{n}+a{x}^{m}+b$;
- L. Jones [79] considered monogenic reciprocal trinomials of type ${x}^{2m}+A{x}^{m}+1$;
- L. Jones [80] showed that there are infinitely many primes p, such that ${x}^{6}+p{x}^{3}+1$ is monogenic with Galois group ${D}_{6}$;
- L. Jones [81] showed that ${x}^{n}+x+1$ is monogenic, if and only if its discriminant is squarefree;
- L. Jones and T. Phillips [82] showed that ${x}^{n}+ax+b$ is monogenic infinitely often;
- L. Jones and D. White [83] found new infinite families of monogenic trinomials of type ${x}^{n}+A{x}^{m}+B$.

**Theorem**

**8**

**.**

- T. A. Gassert, H. Smith and K. E. Stange [84]: ${x}^{4}-6{x}^{2}-kx-3$;
- H. Ben Yakkou [85]: ${x}^{4}+a{x}^{3}+bx+c$;
- J. Harrington and L. Jones [86] constructed new families of quartic polynomials with various Galois groups, which are monogenic infinitely often;
- A. Jakhar and R. Kalwaniya [87]: ${x}^{6}+a{x}^{m}+bx+c$;
- L. Jones [64]: ${x}^{8}+a{x}^{6}+b{x}^{4}+a{x}^{2}+1$;
- L. Jones [89]: ${x}^{p}-2pt{x}^{p-1}+{p}^{2}{t}^{2}{x}^{p-2}+1$;
- Jakhar, A. S. Kaur and S. Kumar [90]: ${x}^{n}+a{x}^{2}+bx+c$;
- Jakhar, A. S. Kaur and S. Kumar [91]: ${x}^{{p}^{s}}-a{x}^{n}-b{x}^{m}-c$;
- A. Jakhar [92]: ${x}^{n}+a{x}^{n-1}+b{x}^{n-2}+c$;
- L. Jones [93] constructed infinite families of reciprocal monogenic polynomials with prescribed Galois group;
- L. Jones [94] showed that if $4\le n\ge m\ge 0$ and $gcd(n,m)=gcd(n,k)=1$ then ${x}^{n-m}{(x+k)}^{m}+p$ is monogenic for infinitely many primes p;
- L. Jones [95]: ${x}^{n}+A{(Bx+1)}^{m}$;
- L. Jones [96]: ${x}^{n}+t\xb7g\left(x\right)$ with $n>deg\left(g\right)$, when $g\left(x\right)$ is monic and $deg\left(g\right)\in \{2,3,4\}$;
- L. Jones [97] constructed reciprocal monogenic quintinomials of type ${x}^{{2}^{n}}+A{x}^{3\xb7{2}^{n-2}}+B{x}^{{2}^{n-1}}+A{x}^{{2}^{n-2}}+1$;
- L. Jones [98] considered infinite families of monogenic quadrinomials, quintinomials and sextinomials.

#### 3.2. The Relative Case

- M. Sahmoudi and M. E. Charkani [101] considered relative pure cyclic extensions;
- A. Soullami, M. Sahmoudi and O. Boughaleb [102]: ${x}^{{3}^{n}}+a{x}^{{3}^{s}}-b$ over number fields;
- O. Boughaleb, A. Soullami and M. Sahmoudi [103]: ${x}^{{p}^{n}}+a{x}^{{p}^{s}}-b$ over number fields;
- H. Smith [104] studied relative radical extensions;
- S. K. Khanduja and B. Jhorar [105] gave equivalent versions of Dedekind’s criterion in general rings;
- R. Sekigawa [108] constructed an infinite number of cyclic relative extensions of prime degree that are relative monogenic.

#### 3.3. Composite Polynomials

- J. Harrington and L. Jones [109] gave conditions for the monogenity of ${({x}^{m}-b)}^{n}-a$, and the composition of ${x}^{n}-a$ and ${x}^{m}-b$;
- Jakhar, A. R. Kalwaniya and P. Yadav [110] considered monogenity of ${({x}^{m}-b)}^{n}-a$, and the composition of ${x}^{n}-a$ and ${x}^{m}-b$ using a refined version of the Dedekind criterion;
- J. Harrington and L. Jones [111] considered monogenity of ${\mathsf{\Phi}}_{{p}^{a}}\left({\mathsf{\Phi}}_{{2}^{b}}\left(x\right)\right)$, where ${\mathsf{\Phi}}_{N}\left(x\right)$ is the cyclotomic polynomial of index N;
- L. Jones [112] considered monotonically stable polynomials of type $g\left({f}^{n}\left(x\right)\right)$;
- L. Jones [113] constructed infinite collections of monic Eisenstein polynomials $f\left(x\right)$, such that $f\left({x}^{{d}^{n}}\right)$ are monogenic for all integers $n\ge 0$ and $d>1$;
- L. Jones [114] considered monogenity of ${S}_{k}\left({x}^{p}\right)$, where ${S}_{k}\left(x\right)={x}^{3}-k{x}^{2}-(k+3)x-1$ the Shanks polynomial;
- L. Jones [115] considered monogenity of $f\left({x}^{p}\right)$, where $f\left(x\right)$ is the characteristic polynomial of an Nth order linear recurrence;
- J. Harrington and L. Jones [116] gave conditions for the monogenity of $f\left({x}^{{p}^{n}}\right)$, where $f\left(x\right)={x}^{m}+a{x}^{m-1}+b$;
- S. Kaur, S. Kumar and L. Remete [117] considered monogenity of $f\left({x}^{k}\right)$, where $f\left(x\right)={x}^{d}+A\xb7h\left(x\right),degh<d$.

**Theorem**

**9**

**.**Let a and b be positive integers, and let p be a prime. Then the polynomial ${\mathsf{\Phi}}_{{p}^{a}}\left({\mathsf{\Phi}}_{{2}^{b}}\left(x\right)\right)$ is monogenic, where ${\mathsf{\Phi}}_{N}\left(x\right)$ is the cyclotomic polynomial of index N.

#### 3.4. Connection with primes

**Theorem**

**10**

**.**

#### 3.5. Number of Generators of Power Integral Bases

- M. Kang and D. Kim [122] considered the number of monogenic orders in pure cubic fields;
- J. H. Evertse [123] considered “rationally monogenic” orders of number fields;
- S. Akhtari [124] showed that a positive proportion of cubic number fields, when ordered by their discriminant, are not monogenic;
- L. Alpöge, M. Bhargava, A. Shnidman [125] showed that, if isomorphism classes of cubic fields are ordered by absolute discriminant, then a positive proportion are not monogenic and yet have no local obstruction to being monogenic (that is, the index form equations represent $+1$ or $-1$ mod p for all primes p);
- M. Bhargava [126] proved that an order O in a quartic number field can have at most 2760 inequivalent generators of power integral bases (and at most 182 if $\left|D\right(O\left)\right|$ is sufficiently large). The problem is reduced to counting the number solutions of cubic and quartic Thue equations, somewhat analogously like described in Section 2.4, using a refined enumeration;
- S. Akhtari [127] gave another proof of Bhargava’s result [126]: she used the more direct approach of Section 2.4 and applied sharp bounds for the numbers of solutions of cubic and quartic Thue equations.

#### 3.6. Miscellaneous

- H. H. Kim [128] showed that the number of monogenic dihedral quartic extensions with absolute discriminant $\le X$ is of size $O\left({X}^{3/4}{(logX)}^{3}\right)$;
- N. Khan, S. Katayama, T. Nakahara and T. Uehara [129] proved that the composite of a totally real field with a cyclotomic field of odd conductor $\ge 3$ or even ≥8 has no power integral basis;
- N. Khan, T. Nakahara and H. Sekiguchi [130] proved that there are exactly three monogenic cyclic sextic fields of prime-power conductor, namely $\mathbb{Q}\left({\zeta}_{7}\right),\mathbb{Q}\left({\zeta}_{9}\right)$ and the maximal real subfield of $\mathbb{Q}\left({\zeta}_{13}\right)$;
- D. Gil-Muňoz and M. Tinková [131] considered the indices of non-monogenic simplest cubic polynomials;
- L. Jones [132] considered infinite families of monogenic Pisot (anti-Pisot) polynomials;
- A. Jakhar and S. K. Khanduja [133] gave lower bounds for the p-index of a polynomial;
- M. Castillo, [134] showed, e.g., that $\mathbb{Q}\left({\alpha}_{n}\right),n\ge 1$ is monogenic, where ${\alpha}_{0}=1$ and ${\alpha}_{n}=\sqrt{2+{\alpha}_{n-1}}$ for $n\ge 1$;
- T. Kashio and R. Sekigawa [135] showed that a monogenic normal cubic field is a simplest cubic field for some parameter;
- F. E. Tanoé [136] considered monogenity of biquadratic fields using a special integer basis;
- Aruna C. and P. Vanchinathan [140] showed that an infinite number of so-called exceptional quartic fields are monogenic.

#### 3.7. Explicit Calculations, Algorithms

- Z. Franŭsić and B. Jadrijević [141] calculated generators of relative power integral bases in a family of quartic extensions of imaginary quadratic fields;
- I. Gaál [142] showed that index form equations in composites of a totally real cubic field and a complex quadratic field can be reduced to absolute Thue equations;
- I. Gaál [143] showed that the index form equations in composites of a totally real field and a complex quadratic field can be reduced to the absolute index form equations of the totally real field;
- I. Gaál [144] considered generators of power integral bases in fields generated by monogenic trinomials of type ${x}^{6}+3{x}^{3}+3a$;
- I. Gaál [145] considered generators of power integral bases in fields generated by monogenic binomial compositions of type ${({x}^{3}-b)}^{2}+1$;
- I. Gaál [146] gave an efficient method to determine all generators of power integral bases of pure sextic fields;
- I. Gaál and L. Remete [147] considered monogenity in octic fields of type $K=\mathbb{Q}\left(\sqrt[4]{a+bi}\right)$;
- I. Gaál [148] determined “small” solutions of the index form equation in $K=\mathbb{Q}\left(\sqrt[6]{m}\right)$, for $-5000<m<0$, such that ${x}^{6}-m$ is monogenic (1521 fields). Experience: $\sqrt[6]{m}$ is the only generator of power integral bases;
- I. Gaál [149] determined “small” solutions of index form equations in $K=\mathbb{Q}\left(\sqrt[8]{m}\right)$, $-5000<m<0$, such that ${x}^{8}-m$ is monogenic (2024 fields). Experience: $\sqrt[8]{m}$ is the only generator of power integral bases, except for $m=-1$;

**Theorem**

**11**

**.**

**Theorem**

**12**

**.**

## 4. Further Research

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Gaál, I.
Monogenity and Power Integral Bases: Recent Developments. *Axioms* **2024**, *13*, 429.
https://doi.org/10.3390/axioms13070429

**AMA Style**

Gaál I.
Monogenity and Power Integral Bases: Recent Developments. *Axioms*. 2024; 13(7):429.
https://doi.org/10.3390/axioms13070429

**Chicago/Turabian Style**

Gaál, István.
2024. "Monogenity and Power Integral Bases: Recent Developments" *Axioms* 13, no. 7: 429.
https://doi.org/10.3390/axioms13070429