Shanks' Cubic Polynomial: Simplest Cubic Fields

• Shanks' cubic polynomial is a one-parameter family defining simplest cyclic cubic fields with discriminant (n²+3n+9)².
• Its symmetry and explicit Gaussian period formulas facilitate concrete analysis of units, ramification, and monogenicity.
• The polynomial extends to function-field analogues and Diophantine equations, offering practical insights in Galois theory and computational number theory.

Shanks’ cubic polynomial is the one-parameter family

$f_n(X)=X^3-nX^2-(n+3)X-1,$

usually considered for $n\in \mathbb Z$ and, in more recent work, also for $n\in \mathbb Q$. For integral $n$, $f_n$ is irreducible over $\mathbb Q$, and a root generates a cyclic cubic field $L_n$ with discriminant $(n^2+3n+9)^2$; these are the classical simplest cubic fields introduced by Shanks in 1974. The family is also a generic cyclic cubic polynomial in the sense that every cyclic cubic field is isomorphic to some $L_n$. Because of this parametrizing role, Shanks’ cubic polynomial lies at the intersection of explicit Galois theory, Gaussian periods, associated orders, normal integral bases, monogenicity, Jacobi–Perron expansions, and several Diophantine constructions (Aoki, 14 Sep 2025).

1. Definition, normalizations, and basic algebraic structure

In the standard number-field normalization, Shanks’ cubic polynomial is

$f_n(X)=X^3-nX^2-(n+3)X-1.$

If $n\in \mathbb Z$0 is a root, then $n\in \mathbb Z$1 is a cyclic cubic field. The family has the symmetry

$n\in \mathbb Z$2

hence $n\in \mathbb Z$3. The Galois group is generated by the automorphism $n\in \mathbb Z$4 given on the root by

$n\in \mathbb Z$5

and one writes $n\in \mathbb Z$6, $n\in \mathbb Z$7 (Ogawa et al., 2023).

A common alternative notation replaces $n\in \mathbb Z$8 by $n\in \mathbb Z$9, writing

$n\in \mathbb Q$0

For $n\in \mathbb Q$1, the corresponding simplest cubic field $n\in \mathbb Q$2 is totally real, and the three real roots satisfy

$n\in \mathbb Q$3

In this form the family is especially convenient for studying units, signatures, and additive indecomposables (Kala et al., 2023).

A further normalization occurs in the theory of Thue equations: $n\in \mathbb Q$4 Then $n\in \mathbb Q$5 is again the defining cubic of a simplest cubic field, so this is a shifted presentation of the same arithmetic family rather than a different class of fields (Levesque et al., 2015).

2. Cyclic cubic parametrization, discriminants, conductors, and monogenicity

For integral $n\in \mathbb Q$6, Shanks’ polynomial is always irreducible over $n\in \mathbb Q$7, and its discriminant is

$n\in \mathbb Q$8

Hence the field $n\in \mathbb Q$9 has discriminant

$n$0

and is Galois over $n$1 with $n$2. More generally, if $n$3 with $n$4, one sets

$n$5

In the cyclic cases considered in the modern ramification theory of the family, one has

$n$6

and an irreducibility criterion suited to the wild case is: if $n$7, $n$8, and $n$9 is square-free, then $f_n$0 is irreducible over $f_n$1 (Aoki, 14 Sep 2025).

The family is generic for cyclic cubic extensions: for any cyclic cubic field $f_n$2, there exists $f_n$3 with $f_n$4. This gives Shanks’ polynomial a universal status among cyclic cubic fields, not merely among special examples (Aoki, 14 Sep 2025).

The behavior at the prime $f_n$5 governs tame versus wild ramification. For simplest cubic fields $f_n$6, tameness is equivalent to $f_n$7 or $f_n$8; otherwise the extension is wildly ramified at $f_n$9. This distinction is reflected both in the conductor and in the structure of the associated order (Hashimoto et al., 2021).

Monogenicity is unusually explicit in this family. Let

$\mathbb Q$0

A cyclic cubic field is monogenic only if it is a simplest cubic field $\mathbb Q$1, and $\mathbb Q$2 is monogenic if and only if

$\mathbb Q$3

Equivalently,

$\mathbb Q$4

When these conditions hold, Kashio and Sekigawa give an explicit power integral basis in terms of $\mathbb Q$5 and $\mathbb Q$6 (Kashio et al., 2019).

3. Gaussian periods, normal integral bases, and associated orders

For a cyclic cubic field $\mathbb Q$7 with conductor $\mathbb Q$8, let $\mathbb Q$9. The cubic Gaussian periods are

$L_n$0

and the period polynomial is

$L_n$1

Each $L_n$2 generates $L_n$3, and the period polynomial is related to $L_n$4 by an explicit linear change of variable and a scalar factor (Aoki, 14 Sep 2025).

When $L_n$5 is tamely ramified, square-free conductor is equivalent to the existence of a normal integral basis, and Hashimoto–Aoki classify all generators explicitly in terms of roots of $L_n$6. In that case the Gaussian periods themselves generate a normal integral basis, extending earlier formulas of Lehmer, Châtelet, and Lazarus from the case $L_n$7 to all tamely ramified simplest cubic fields (Hashimoto et al., 2021).

The tame and wild period–root relations are parallel but not identical:

Ramification	Linear relation between periods and roots	Associated order
Tame	$L_n$8	$L_n$9
Wild	$(n^2+3n+9)^2$0	$(n^2+3n+9)^2$1

In the wildly ramified case, the module structure is no longer that of a normal integral basis. Writing

$(n^2+3n+9)^2$2

one has

$(n^2+3n+9)^2$3

The wild associated order has a larger unit group than in the tame case; in particular, it contains units involving the idempotent of the trivial character, and this enlargement is crucial in proving that the Gaussian periods are, up to the sign $(n^2+3n+9)^2$4, exactly the conjugates of $(n^2+3n+9)^2$5 (Ogawa et al., 2023).

4. Units, indecomposables, norm restrictions, and Jacobi–Perron structure

In simplest cubic fields $(n^2+3n+9)^2$6 with defining polynomial

$(n^2+3n+9)^2$7

the unit rank is $(n^2+3n+9)^2$8, $(n^2+3n+9)^2$9 is a system of fundamental units in $L_n$0, units of all signatures occur, and every totally positive unit is a square. Up to multiplication by totally positive units, the totally positive indecomposable elements are exactly

$L_n$1

and

$L_n$2

Using the notation

$L_n$3

the same work computes a periodic homogeneous Jacobi–Perron expansion for the vector

$L_n$4

proves that every semiconvergent is $L_n$5-indecomposable in its signature, and, for $L_n$6, shows that suitable second-order semiconvergents recover all indecomposables up to units and conjugation. It also records Tinková’s theorem that

$L_n$7

for simplest cubic fields generated by roots of $L_n$8 with $L_n$9 (Kala et al., 2023).

The family also admits sharp norm bounds. Under the squarefree hypothesis

$f_n(X)=X^3-nX^2-(n+3)X-1.$0

so that $f_n(X)=X^3-nX^2-(n+3)X-1.$1, Lemmermeyer and Pethő prove that for every $f_n(X)=X^3-nX^2-(n+3)X-1.$2, either

$f_n(X)=X^3-nX^2-(n+3)X-1.$3

or $f_n(X)=X^3-nX^2-(n+3)X-1.$4 is associated to an integer. Moreover, if

$f_n(X)=X^3-nX^2-(n+3)X-1.$5

then $f_n(X)=X^3-nX^2-(n+3)X-1.$6 is associated to one of the conjugates of $f_n(X)=X^3-nX^2-(n+3)X-1.$7. They use this restriction on principal norms to simplify the construction of unramified biquadratic extensions such as

$f_n(X)=X^3-nX^2-(n+3)X-1.$8

when $f_n(X)=X^3-nX^2-(n+3)X-1.$9 is a square (Lemmermeyer et al., 2012).

5. Ramanujan cubics, Thue equations, and power-compositional variants

Shanks cubic polynomials form the $n\in \mathbb Z$00 subfamily of Ramanujan cubic polynomials

$n\in \mathbb Z$01

For $n\in \mathbb Z$02,

$n\in \mathbb Z$03

and the roots admit an explicit trigonometric formula in terms of

$n\in \mathbb Z$04

The rational map

$n\in \mathbb Z$05

cyclically permutes the roots of a Ramanujan cubic; in the Shanks case this becomes

$n\in \mathbb Z$06

When $n\in \mathbb Z$07 is a prime $n\in \mathbb Z$08, the roots of the Shanks cubic are affine transforms of cubic Gaussian periods, and the resulting identities connect Shanks’ family to Lehmer’s period polynomials (Barbero et al., 2014).

A general normal-form theorem places this within the geometry of all cubics: every monic cubic with distinct roots is either a translation of $n\in \mathbb Z$09 or is linearly conjugate to a Ramanujan simple cubic $n\in \mathbb Z$10. Replacing $n\in \mathbb Z$11 by $n\in \mathbb Z$12 in $n\in \mathbb Z$13 yields the Shanks polynomials, so the Möbius symmetry of simplest cubic fields appears as a special case of a wider order-three transformation theory for cubic roots (Dresden et al., 2017).

Diophantine generalizations preserve the same algebraic core. Thomas’s classical Thue form

$n\in \mathbb Z$14

is associated to the simplest cubic cyclic fields, and Levesque–Waldschmidt extend it to

$n\in \mathbb Z$15

thereby effectively solving the two-parameter family

$n\in \mathbb Z$16

by reducing it to norm equations for powers of units in simplest cubic fields (Levesque et al., 2015).

A different extension studies power-compositional Shanks polynomials

$n\in \mathbb Z$17

If $n\in \mathbb Z$18 and the corresponding

$n\in \mathbb Z$19

is squarefree, then $n\in \mathbb Z$20 is monogenic. For a prime $n\in \mathbb Z$21 such that $n\in \mathbb Z$22 is irreducible in $n\in \mathbb Z$23, $n\in \mathbb Z$24 is a $n\in \mathbb Z$25-Shanks prime if and only if $n\in \mathbb Z$26 is non-monogenic; by contrast, $n\in \mathbb Z$27 is monogenic for every prime divisor $n\in \mathbb Z$28 of $n\in \mathbb Z$29 (Jones, 2023).

6. Function-field analogue and overall mathematical role

Over $n\in \mathbb Z$30 with characteristic greater than five, the Galois simple cubic function fields of unit rank two with $n\in \mathbb Z$31-exceptional units are defined by the same polynomial shape

$n\in \mathbb Z$32

This is the immediate function-field analogue of Shanks’ simplest cubic number fields. In that setting the extension is cyclic of degree $n\in \mathbb Z$33, the discriminant is

$n\in \mathbb Z$34

and, if $n\in \mathbb Z$35 is cube-free, the regulator is

$n\in \mathbb Z$36

with $n\in \mathbb Z$37 forming a fundamental system of units (Rozenhart et al., 2011).

The same explicitness supports class number computations by truncated Euler products. In particular, over $n\in \mathbb Z$38 and $n\in \mathbb Z$39, one obtains a classification of all Galois simple cubic function fields with ideal class number one under the cube-free hypothesis on $n\in \mathbb Z$40 (Rozenhart et al., 2011).

Taken together, these developments show that Shanks’ cubic polynomial is not merely a convenient source of examples. It is a uniform polynomial model for cyclic cubic extensions, one that admits parallel descriptions through roots, Gaussian periods, Dirichlet characters, Galois-module generators, unit equations, and continued-fraction-like algorithms. Its arithmetic remains unusually explicit under tame and wild ramification, under passage to function fields, and under several Diophantine and compositional extensions.