---
title: 'Shanks'' Cubic Polynomial: Simplest Cubic Fields'
url: https://www.emergentmind.com/topics/shanks-cubic-polynomial
type: topic
---

# Shanks' Cubic Polynomial: Simplest Cubic Fields

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 [2509.11137].

## 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 \(\rho_n\) is a root, then \(L_n=\mathbb Q(\rho_n)\) is a cyclic cubic field. The family has the symmetry
\[
f_n(X)=-X^3f_{-n-3}(1/X),
\]
hence \(L_n=L_{-n-3}\). The Galois group is generated by the automorphism \(\sigma\) given on the root by
\[
\sigma(\rho_n)=-\frac{1}{1+\rho_n},
\]
and one writes \(\rho_n'=\sigma(\rho_n)\), \(\rho_n''=\sigma^2(\rho_n)\) [2305.08888].

A common alternative notation replaces \(n\) by \(a\), writing
\[
x^3-ax^2-(a+3)x-1.
\]
For \(a\ge -1\), the corresponding simplest cubic field \(K=\mathbb Q(\rho)\) is totally real, and the three real roots satisfy
\[
a+1<\rho,\qquad -2<\rho'<-1,\qquad -1<\rho''<0.
\]
In this form the family is especially convenient for studying units, signatures, and additive indecomposables [2303.00485].

A further normalization occurs in the theory of Thue equations:
\[
F_n(X,Y)=X^3-(n-1)X^2Y-(n+2)XY^2-Y^3.
\]
Then \(F_n(X,1)\) 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 [1505.06708].

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

For integral \(n\), Shanks’ polynomial is always irreducible over \(\mathbb Q\), and its discriminant is
\[
d(f_n)=(n^2+3n+9)^2.
\]
Hence the field \(L_n\) has discriminant
\[
D_{L_n}=(n^2+3n+9)^2,
\]
and is Galois over \(\mathbb Q\) with \(\operatorname{Gal}(L_n/\mathbb Q)\simeq C_3\). More generally, if \(n=n_1/n_2\) with \((n_1,n_2)=1\), one sets
\[
\Delta_n=n_1^2+3n_1n_2+9n_2^2.
\]
In the cyclic cases considered in the modern ramification theory of the family, one has
\[
D_{L_n}=\Delta_n^2,\qquad \mathfrak f=\Delta_n,
\]
and an irreducibility criterion suited to the wild case is: if \(3\mid n_1\), \(9\parallel \Delta_n\), and \(\Delta_n/9\) is square-free, then \(f_n(X)\) is irreducible over \(\mathbb Q\) [2509.11137].

The family is generic for cyclic cubic extensions: for any cyclic cubic field \(L\), there exists \(n\in\mathbb Q\) with \(L\simeq L_n\). This gives Shanks’ polynomial a universal status among cyclic cubic fields, not merely among special examples [2509.11137].

The behavior at the prime \(3\) governs tame versus wild ramification. For simplest cubic fields \(L_n/\mathbb Q\), tameness is equivalent to \(3\nmid n\) or \(n\equiv 12\pmod{27}\); otherwise the extension is wildly ramified at \(3\). This distinction is reflected both in the conductor and in the structure of the associated order [2108.03367].

Monogenicity is unusually explicit in this family. Let
\[
\Delta_t=t^2+3t+9,\qquad K_t=\mathbb Q(\theta_t),\qquad f_t(x)=x^3-tx^2-(t+3)x-1.
\]
A cyclic cubic field is monogenic only if it is a simplest cubic field \(K_t\), and \(K_t\) is monogenic if and only if
\[
\frac{\Delta_t}{c_{K_t}}\in\mathbb N^3.
\]
Equivalently,
\[
t\not\equiv 3,21 \pmod{27},\qquad v_p(\Delta_t)\not\equiv 2\pmod 3\quad\text{for all }p\neq 3.
\]
When these conditions hold, Kashio and Sekigawa give an explicit power integral basis in terms of \(\theta_t\) and \(\Delta_t/c_{K_t}\) [1912.03103].

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

For a cyclic cubic field \(L\) with conductor \(\mathfrak f\), let \(\zeta_{\mathfrak f}=e^{2\pi i/\mathfrak f}\). The cubic Gaussian periods are
\[
\eta_0=\operatorname{Tr}_{\mathbb Q(\zeta_{\mathfrak f})/L}(\zeta_{\mathfrak f}),\qquad
\eta_1=\sigma(\eta_0),\qquad
\eta_2=\sigma^2(\eta_0),
\]
and the period polynomial is
\[
P(X)=(X-\eta_0)(X-\eta_1)(X-\eta_2)\in\mathbb Z[X].
\]
Each \(\eta_i\) generates \(L\), and the period polynomial is related to \(f_n\) by an explicit linear change of variable and a scalar factor [2509.11137].

When \(L_n/\mathbb Q\) 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 \(f_n\). In that case the Gaussian periods themselves generate a normal integral basis, extending earlier formulas of Lehmer, Châtelet, and Lazarus from the case \(f_{L_n}=n^2+3n+9\) to all tamely ramified simplest cubic fields [2108.03367].

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

| Ramification | Linear relation between periods and roots | Associated order |
|---|---|---|
| Tame | \(\eta_i=\mu(\mathfrak f)\left(n_2\rho_n^{(\sigma^i)}+\frac{1-n_1}{3}\right)\) | \(\mathcal A_{L/\mathbb Q}=\mathbb Z[G]\) |
| Wild | \(\eta_i=\mu(\mathfrak f/9)\left(n_2\rho_n^{(\sigma^i)}-\frac{n_1}{3}\right)\) | \(\mathcal A_{L/\mathbb Q}=\mathbb Z[G][{\bf e}_{\mathfrak f},{\bf e}_{\mathfrak f/3}]\) |

In the wildly ramified case, the module structure is no longer that of a normal integral basis. Writing
\[
\alpha=n_2\rho_n-\frac{n_1}{3},
\]
one has
\[
\mathcal O_{L_n}=\mathcal A_{L_n/\mathbb Q}(\alpha+1).
\]
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 \(\mu(\mathfrak f/9)\), exactly the conjugates of \(\alpha\) [2305.08888].

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

In simplest cubic fields \(K=\mathbb Q(\rho)\) with defining polynomial
\[
x^3-ax^2-(a+3)x-1,
\]
the unit rank is \(2\), \(\{\rho,\rho'\}\) is a system of fundamental units in \(\mathbb Z[\rho]\), 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
\[
1,\qquad 1+\rho+\rho^2,
\]
and
\[
\alpha_{v,w}=-v-(v(a+2)+1+w)\rho+(v+1)\rho^2,
\qquad 0\le v\le a,\quad 0\le w\le a-v.
\]
Using the notation
\[
Q_{v,w}=-v-(v(a+2)+1+w)\rho+(v+1)\rho^2,
\]
the same work computes a periodic homogeneous Jacobi–Perron expansion for the vector
\[
(1,-\rho',\rho'^2),
\]
proves that every semiconvergent is \(s\)-indecomposable in its signature, and, for \(a\ge 15\), shows that suitable second-order semiconvergents recover all indecomposables up to units and conjugation. It also records Tinková’s theorem that
\[
P(\mathbb Z[\rho])=6
\]
for simplest cubic fields generated by roots of \(x^3-ax^2-(a+3)x-1\) with \(a\ge 3\) [2303.00485].

The family also admits sharp norm bounds. Under the squarefree hypothesis
\[
m=a^2+3a+9,
\]
so that \(\mathcal O_{K_a}=\mathbb Z[\alpha]\), Lemmermeyer and Pethő prove that for every \(y\in\mathbb Z[\alpha]\), either
\[
|N(y)|\ge 2a+3,
\]
or \(y\) is associated to an integer. Moreover, if
\[
|N(y)|=2a+3,
\]
then \(y\) is associated to one of the conjugates of \(\alpha-1\). They use this restriction on principal norms to simplify the construction of unramified biquadratic extensions such as
\[
K\bigl(\sqrt{\alpha+2},\sqrt{\alpha'+2}\bigr)
\]
when \(2a+3\) is a square [1202.6022].

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

Shanks cubic polynomials form the \(s=-1\) subfamily of Ramanujan cubic polynomials
\[
p(h,s,x)=x^3+hsx^2-(h+3)s^2x+s^3.
\]
For \(s=-1\),
\[
p(h,-1,x)=x^3-hx^2-(h+3)x-1,
\]
and the roots admit an explicit trigonometric formula in terms of
\[
T(h)=h^2+3h+9.
\]
The rational map
\[
n_s(z)=\frac{s^2}{s-z}
\]
cyclically permutes the roots of a Ramanujan cubic; in the Shanks case this becomes
\[
n_{-1}(z)=\frac{1}{1+z}.
\]
When \(T(h)\) is a prime \(p\equiv 1\pmod 3\), 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 [1401.1474].

A general normal-form theorem places this within the geometry of all cubics: every monic cubic with distinct roots is either a translation of \(x^3\) or is linearly conjugate to a Ramanujan simple cubic \(p_B(x)\). Replacing \(x\) by \(-x\) in \(p_B\) 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 [1709.00534].

Diophantine generalizations preserve the same algebraic core. Thomas’s classical Thue form
\[
F_n(X,Y)=X^3-(n-1)X^2Y-(n+2)XY^2-Y^3
\]
is associated to the simplest cubic cyclic fields, and Levesque–Waldschmidt extend it to
\[
F_{n,a}(X,Y)=(X-\lambda_0^aY)(X-\lambda_1^aY)(X-\lambda_2^aY),
\]
thereby effectively solving the two-parameter family
\[
F_{n,a}(x,y)=\pm 1
\]
by reducing it to norm equations for powers of units in simplest cubic fields [1505.06708].

A different extension studies power-compositional Shanks polynomials
\[
\mathcal S_k(x)=x^3-kx^2-(k+3)x-1,\qquad \mathcal S_k(x^p).
\]
If \(k\not\equiv 3\pmod 9\) and the corresponding
\[
\mathcal D=
\begin{cases}
(k/3)^2+k/3+1,& k\equiv 0\pmod 3,\\
k^2+3k+9,& \text{otherwise}
\end{cases}
\]
is squarefree, then \(\mathcal S_k(x)\) is monogenic. For a prime \(p\) such that \(\mathcal S_k(x)\) is irreducible in \(\mathbb F_p[x]\), \(p\) is a \(k\)-Shanks prime if and only if \(\mathcal S_k(x^p)\) is non-monogenic; by contrast, \(\mathcal S_k(x^p)\) is monogenic for every prime divisor \(p\) of \(k^2+3k+9\) [2303.11872].

## 6. Function-field analogue and overall mathematical role

Over \(\mathbb F_q(t)\) with characteristic greater than five, the Galois simple cubic function fields of unit rank two with \(k\)-exceptional units are defined by the same polynomial shape
\[
x^3-Ax^2-(A+3)x-1.
\]
This is the immediate function-field analogue of Shanks’ simplest cubic number fields. In that setting the extension is cyclic of degree \(3\), the discriminant is
\[
D(K)=\frac{(A^2+3A+9)^2}{I^2},\qquad I=\gcd(A^2+3A+9,A'),
\]
and, if \(A^2+3A+9\) is cube-free, the regulator is
\[
R=\deg(A)^2,
\]
with \(\{\varepsilon,\varepsilon+1\}\) forming a fundamental system of units [1108.6048].

The same explicitness supports class number computations by truncated Euler products. In particular, over \(\mathbb F_5\) and \(\mathbb F_7\), one obtains a classification of all Galois simple cubic function fields with ideal class number one under the cube-free hypothesis on \(A^2+3A+9\) [1108.6048].

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.

Source: https://www.emergentmind.com/topics/shanks-cubic-polynomial