Monogenity and Power Integral Bases: Recent Developments

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled"Monogenity and Power Integral Bases". We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given.


Introduction
Let K = Q(γ) be an algebraic number field of degree n, generated by the algebraic integer γ, with ring of integers Z K and discriminant D K .It is a classical problem of algebraic number theory, going back to R. Dedekind [1], K. Hensel [2] and H. Hasse [3] to decide if the ring Z K can be generated by a single element α ∈ Z K , that is, if it is mono-generated, Z K = Z[α].In this case, we say that the ring Z K , or the field K, is monogenic, and the integral basis {1, α, . . .α n−1 } is called a power integral basis.
Recently, this area has been developing very rapidly.In order to create a suitable forum to present recent results on monogenity, the author started a series of online meetings "Monogenity and Power Integral Bases" (https://sway.cloud.microsoft/F2kZzeZ3bmD4dFfy?ref=Link accessed on 15 January 2021) in 2021.The purpose was to make contacts, circulate preprints and results, and support collaboration between researchers all over the world working in this area.During the time of pandemy this was the only way to contact, but later on this proved to be an easy and fast way of contacting.Therefore, up to March 2024 we already had nine meetings and we hope to continue.
The main purpose of this paper is to give an overview of the latest developments in monogenity theory, about the results that were presented at the online meetings and the results that appeared parallel.The paper is also a kind of extension of the book [16] that appeared in 2019.Most of these results are not yet contained there.
In Section 2, we collect the most important tools that were used in several works.These may be useful for further applications.Section 3 collects the most important results, and finally in Section 4 we try to indicate some possible perspectives of further research.
In favour of the reader, we collect some further concepts on monogenity.
For any primitive element α ∈ Z K (that is K = Q(α)), the index of α is defined as the module index We obviously have where D(α) is the discriminant of α, α (i) denoting the conjugates of α corresponding to γ (i) (i = 1, . . ., n) (in the following we shall denote similarly the conjugates of any element of K).Obviously, } is an integral basis, or in other words α generates a power integral basis in K.
The field index, m(K), of K is defined as If K is monogenic, there are elements of index 1, and the field index is also equal to 1.The converse is not true: the field index may happen to be 1 without the field being monogenic.
If α, β are primitive elements in Z K and α + β ∈ Z or α − β ∈ Z then obviously their indices are equal.Such elements are called equivalent.It was proved by B. J. Birch and J. R. Merriman [4] and then in an effective form by K. Győry [5] that up to equivalence there are only finitely many generators of power integral bases in any number field K.
For any integral basis (1, ω 2 , . . ., Then (see [16]) where I(X 2 , . . ., X n ) is a homogeneous polynomial of degree n(n − 1)/2 with integer coefficients, with the property that for any primitive element α = x 1 + ω 2 x 2 + . . .ω n x n ∈ Z K we have The polynomial I(X 2 , . . ., X n ) is called the index form corresponding to the integral basis (1, ω 2 , . . ., ω n ).Since equivalent algebraic integers have the same index, it is independent of X 1 .Therefore, determining elements α ∈ Z K of index m is equivalent to solving the index form equation I(x 2 , . . . ,x n ) = m in x 2 , . . ., x n ∈ Z.
A non-zero irreducible polynomial f (x) ∈ Z[x] is called monogenic if a root α of f (x) generates a power integral basis in the field K = Q(α).Obviously, if the polynomial f (x) is monogenic, then K is also monogenic, but the converse is not true.The field K may happen to be monogenic without f (x) being monogenic.The index of f (x) is defined as ind( f ) = (Z K : Z[α]).

Tools 2.1. Dedekind's Criterion
Let f (x) = ∏ r i=1 ϕ i (x) ℓ i modulo p be the factorization of f (x) modulo p into powers of monic irreducible coprime polynomials of F p [x].
For completeness, we recall here a well-known theorem of Dedekind: Theorem 1 (Chapter I, Proposition 8.3 of [6]).If p does not divide the index where p i = pZ K + ϕ i (α)Z K and the residue degree of p i is f (p i ) = deg(ϕ i ).
As indicated above, it is very important to have a tool to determine prime divisors of the indices of algebraic integers.Therefore, the following well-known criterion of Dedekind is very frequently used: Theorem 2 (Dedekind's criterion [1], see also [7] Theorem 6.1.4,[8] p. 295).Let f (x) ∈ Z[x] be a monic non-zero irreducible polynomial with a root α, let K = Q(α), and let p be a prime number.Let f (x) = ∏ r i=1 ϕ i (x) ℓ i mod p be the factorization of f (x) in F p [x], with monic ϕ i ∈ Z[x], such that their reductions ϕ i (x) are irreducible and pairwise coprime over F p .Set p .
Then M(x) ∈ Z[x] and the following statements are equivalent: 1. p does not divide the index I(α) = (Z K : Z[α]).

The Field Index
We also recall a simple but very important statement of Hensel: Theorem 3 (K.Hensel [2] p. 280).
The prime factors of the field index are smaller than the degree of the field.
Denote by ν p (k) the highest power of the prime p dividing the integer k.
For number fields of degree n ≤ 7, ν p (m(K)) is explicitly determined by the factorization of p into powers of prime ideals of pZ K .
The corresponding tables of [9] are too long to include here, but they present the explicit exponents.

Newton Polygon Method
If p divides the index I(α) = (Z K : Z[α]) then Dedekind's Theorem 1 cannot be applied.
Using Newton polygons, an alternative method was given by Ore [10] to calculate I(α) = (Z K : Z[α]), D K and the prime ideal factorization of primes in Z K .This was further developed among others by J. Montes and E. Nart [11], Fadil, L.E.J. Montes and E. Nart [12] and L. El Fadil [13].This theory was extended to so-called higher-order Newton polygons by J. Guardia, J. Montes and E. Nart [14].The method is also called Montes algorithm.
Here we only give a short introduction to some basic notions and statement of this very technical method, based on the explanation used in [15].During recent years, a huge amount of papers have applied this method.
For any prime p, let ν p be the p-adic valuation of Q. Denote by Q p its p-adic completion and by Z p the ring of p-adic integers.Let ν p be the Gauss's extension of ν p to Q p (x), ν p (P) = min(ν p (a i ), (i = 0, . . ., n) for any polynomial , and let F ϕ be the field F p [x]/(ϕ).For any monic polynomial f (x) ∈ Z p [x], upon the Euclidean division by successive powers of ϕ, we expand f (x) as follows: ).The ϕ-Newton polygon of f (x) with respect to p is the lower boundary convex envelope of the set of points {(i, ν p (a i (x))), a i (x) ̸ = 0} in the Euclidean plane, which we denote by N ϕ f .The ϕ-Newton polygon of f is the process of joining the edges S 1 , . . ., S r ordered by increasing slopes, which can be expressed as For every side, S i , of N ϕ f , the length of S i , denoted ℓ(S i ), is the length of its projection to the x-axis.Its height, denoted by h(S i ), is the length of its projection to the y-axis.Let d(S i ) = gcd(ℓ(S i ), h(S i )) be the ramification degree of S. The principal ϕ-Newton polygon of f , denoted N + ϕ f , is the part of the polygon N ϕ f , which is determined by joining all sides of negative slopes.For every side, S, of N + ϕ f , with initial point (s, u s ) and length ℓ, and for every 0 ≤ i ≤ ℓ, we attach the residue coefficient c i ∈ F ϕ as follows: where (p, ϕ(x)) is the maximal ideal of Z p [x] generated by p and ϕ.Let λ = −h/e be the slope of S, where h and e are two positive coprime integers.Then d = ℓ/e is the degree of S.
The points with integer coordinates lying on S are exactly Thus, if i is not a multiple of e, then (s + i, u s+i ) does not lie in S, and so is called the residual polynomial of f (x) associated to the side S, where, for every i = 0, . . ., d, be the principal ϕ-Newton polygon of f with respect to p.We say that f is a ϕ-regular polynomial with respect to p, if f S i (y) is square free in F ϕ [y] for every i = 1, . . ., r.The polynomial f is said to be p-regular if f (x) = ∏ r i=1 ϕ i (x) ℓ i for some monic polynomials ϕ 1 , . . ., ϕ t of Z[x], such that ϕ 1 , . . ., ϕ t are irreducible coprime polynomials over F p and f is a ϕ i -regular polynomial with respect to p for every i = 1, . . ., t. [12]), denoted by ind ϕ ( f ), is deg(ϕ) times the number of points with natural integer coordinates that lie below or on the polygon N + ϕ f , strictly above the horizontal axis and strictly beyond the vertical axis (see Figure 1).In the example of Figure 1, , which are irreducible and pairwise coprime in F p [x] (i = 1, . . ., r).
For every i = 1, . . ., r, let N + ϕ i ( f ) = S i1 + • • • + S ir i be the principal ϕ i -Newton polygon of f with respect to p.For every j = 1, . . ., r i , let Then we have the following index theorem of Ore.

1.
We have The equality holds if f (x) is p-regular.2.
If f (x) is p-regular, then is the factorization of pZ K into powers of prime ideals of Z K lying above p, where e ij = ℓ ij /d ij , ℓ ij is the length of S ij , d ij is the ramification degree of S ij , and is the residue degree of the prime ideal p ijk over p.

Algorithmic Methods
Several of known efficient methods for the resolutions of Diophantine equations are related to Thue equations, cf.[16].These methods are implemented, e.g., in Magma [17].Therefore, the most efficient methods for solving index form equations also reduce the index form equation to Thue equations.
In cubic fields, the index form equation is a cubic Thue equation, see [16].
The below method of I. Gaál, A. Pethő and M. Pohst [18,19] reduces the index form equations in quartic fields to a cubic and some corresponding quartic Thue equations.This method is quite often used even nowadays, and therefore we briefly present it.
Let K = Q(ξ) be a quartic number field and f (x with a α , x, y, z ∈ Z, and with a common denominator d ∈ Z.Consider the solutions of the equation for 0 < m ∈ Z.We have Theorem 6 ([18]).
Let i m = d 6 m/n, where n = I(ξ).The element α of ( 1) is a solution of (2), if and only if there is a solution (u, v) ∈ Z 2 of the cubic equation such that (x, y, z) satisfies Equation ( 3) is either trivial to solve (when F is reducible), or it is a cubic Thue equation.
For a solution (u, v) of (3), we set Q 0 (x, y, z) = uQ 2 (x, y, z) − vQ 1 (x, y, z).If α in ( 1) is a solution of (2), then If (x 0 , y 0 , z 0 ) ∈ Z 3 is a non-trivial solution of ( 5), with, say, z 0 ̸ = 0 (such a solution can be easily found, see L. J. Mordell [20]), then we can parametrize the solutions x, y, z in the form with rational parameters r, p, q.Substituting these x, y, z into (5), we obtain an equation of the form r(c with integer coefficients c 1 , . . ., c 5 .Multiply the equations in ( 6) by c 1 p + c 2 q and replace r(c 1 p + c 2 q) by c 3 p 2 + c 4 pq + c 5 q 2 .Further multiply the equations in ( 6) by the square of the common denominator of p, q to obtain all integer relations (cf.[19]).We divide those by gcd(p, q) 2 and obtain kx = c 11 p 2 + c 12 pq + c 13 q 2 , ky = c 21 p 2 + c 22 pq + c 23 q 2 , kz = c 31 p 2 + c 32 pq + c 33 q 2 , ( 7) with integer c ij and integer parameters p, q.Here, k is an integer parameter with the property that k divides the det(C)/d 2 0 , where C is the 3 × 3 matrix with entries c ij and d 0 is the gcd of its entries (cf.[19]).Finally, substituting the x, y, z in ( 7) into ( 4) we obtain According to [19], at least one of the equations in Equation ( 8) is a quartic Thue equation over the original number field K.

Pure Fields, Trinomials, Quadrinomials, etc.
There is no doubt that the Newton polygon method has been the most powerful tool during the last couple of years.It is frequently combined with the application of Dedekind's criterion.While, in 2014, S. Ahmad, T. Nakahara and M. Syed [21] investigated monogenity properties of pure sextic fields using their subfield structure and relative monogenity, in 2017 T. A. Gassert [22] already used Montes algorithm to describe monogenity of pure fields.Note that this is only about the monogenity of the polynomials and not the monogenity of number fields generated by a root of the polynomial (for some corrections, see L. El Fadil [23]).
Together with Newton polygons (or instead of them), Dedekind's criterion and Engström's theorem are also often used.The following results often deal with polynomials of similar shape.It is important to add that, especially using Newton polygons, the whole calculation must be performed separately, even for polynomials of similar shape.
The first results investigated monogenity in pure fields (or radical extensions) generated by a root of an irreducible binomial of type x n − m.Assuming that m is squarefree, conditions were given for the monogenity (or non-monogenity) of such pure fields, for n = 6, 8, 12, . .., etc.A following step was to consider general exponents like n = 2 k , 2 k • 3 ℓ , . .., etc., and later on n = p k with a prime p.For some exponents, the more complicated case of a composite m was also investigated.Here is a list of such results, for brevity indicating only the exponents considered: • 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; Theorem 9 (J.Harrington and L. Jones [111]).Let a and b be positive integers, and let p be a prime.Then the polynomial Φ p a (Φ 2 b (x)) is monogenic, where Φ N (x) is the cyclotomic polynomial of index N.

Connection with primes
L. Jones [118][119][120], and J. Harrington and L. Jones [121] detected relations of monogenity of power compositional polynomials with properties of primes.We present here one of these statements.
. Suppose that k ̸ ≡ 0 (mod 4) and that D is squarefree.Let h denote the class number of Q( √ D).Let s ≥ 1 be an integer, such that, for every odd prime divisor p of s, D is not a square modulo p and gcd(p, hD) = 1.Then x 2s n − kx s n − 1 is monogenic for all integers n ≥ 1, if and only if no prime divisor of s is a k-Wall-Sun-Sun prime.

Number of Generators of Power Integral Bases
Some further results considered the number of non-equivalent 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 |D(O)| 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.

Miscellaneous
In addition to the above lists, there were several further interesting statements achieved for monogenity.We try to recall them here.

Explicit Calculations, Algorithms
The powerful methods of Dedekind's criterion and Newton polygons often decide about the monogenity of number fields.However, to explicitly determine all inequivalent generators of power integral bases one needs to perform calculations.These algorithms usually involve Baker-type estimates, reduction methods and enumeration algorithms, cf.[16].There are efficient algorithms for low degree fields and some more complicated methods for higher degree fields.Since these procedures usually require considerable CPU time, if the number field is of high degree, or we need information about a large number of fields, then we turn to the so-called "fast" algorithms for determining "small" solutions.This yields a fast method to determine solutions of the index form equation with absolute values, say ≤10 100 .These algorithms determine all solutions with a high probability but do not exclude extremely large solutions (which, however, nobody has ever met).
We collect here some recent results involving explicit determination of generators of power integral bases.

•
Z. Fran ȗsić 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 + 3x 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 = Q( 4 √ a + bi); • I. Gaál [148] determined "small" solutions of the index form equation in K = Q( 6 √ m), for −5000 < m < 0, such that x 6 − m is monogenic (1521 fields).Experience: 6  √ m is the only generator of power integral bases; • I. Gaál [149] determined "small" solutions of index form equations in K = Q( 8 √ m), −5000 < m < 0, such that x 8 − m is monogenic (2024 fields).Experience: 8  √ m is the only generator of power integral bases, except for m = −1; • I. Gaál [150] extended [46] on monogenity properties of trinomials of type x 4 + ax 2 + b; • I. Gaál [151] calculated generators of power integral bases in families of number fields generated by a root of monogenic quartic polynomials considered in [86].