Norm Forms

Summary: Homogeneous polynomial forms in $n$ variables obtained as the symmetric product ${\prod }_i(x_0+{\alpha }_i{x}_1+{\alpha }_i^2{x}_2+\cdots +{\alpha }_i^{n-1}{x}_{n-1})$ over the roots of a degree-$n$ polynomial. The norm form is multiplicatively closed: products and powers of two such forms are again of the same shape.

Sources: additions-9 (Articles 89–94).

Last updated: 2026-05-10.

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Definition

Let $f(s)=s^n-a_1{s}^{n-1}+a_2{s}^{n-2}-\cdots \pm a_n=0$ have roots ${\alpha }_1,\dots ,{\alpha }_n$. The norm form in indeterminates $x_0,\dots ,x_{n-1}$ is $N_f(x_0,\dots ,x_{n-1})={\prod }_{i=1}^n(x_0+{\alpha }_i{x}_1+{\alpha }_i^2{x}_2+\cdots +{\alpha }_i^{n-1}{x}_{n-1}).$ Although the individual factors are irrational (involving the roots ${\alpha }_i$), their product is symmetric in the roots and hence rational — its coefficients are polynomials in the elementary symmetric functions $a_1,\dots ,a_n$ of $f$ (source: additions-9, Art. 89).

In modern language, $N_f$ is the field norm $N_{K/\mathbb{Q}}$ on the algebraic number field $K=\mathbb{Q}[s]/(f(s))$, expressed in the power-basis $1,\alpha ,{\alpha }^2,\dots ,{\alpha }^{n-1}$.

Cases worked out by Lagrange

Quadratic ($n=2$, $f=s^2-as+b$): $N_f(x,y)=x^2+axy+b{y}^2.$ With $a=0$ this reduces to $x^2+b{y}^2$, the form treated in ch2.0.11-quadratic-form-factorization.

Cubic ($n=3$, $f=s^3-a{s}^2+bs-c$): $N_f(x,y,z)=x^3+a{x}^{2}y+bx{y}^2+c{y}^3+(a^2-2b)x^{2}z+(ab-3c)xyz+(b^2-2ac)x{z}^2+ac{y}^{2}z+bcy{z}^2+c^2{z}^3.$ Ten terms, derived from Newton’s identities for symmetric power sums of the three roots.

Quartic ($n=4$): sketched in additions-9, Art. 94 via the elimination $\rho =x+sy+s^{2}z+s^{3}t$ from $f(s)=0$, which yields a quartic in $\rho$ whose constant term is $N_f$.

Multiplicativity

The fundamental property (additions-9, Arts. 89, 91): $N_f(\mathbf{x})\cdot N_f({\mathbf{x}}^{\prime })=N_f(\mathbf{X})$ where $\mathbf{X}=\mathbf{x}\star {\mathbf{x}}^{\prime }$ is computed component-wise from the partial product $(x_0+\alpha x_1+\cdots )(x_0^{\prime }+\alpha x_1^{\prime }+\cdots )$ expanded and reduced modulo $f(\alpha )=0$. The result is the composition law, and is invariant under the choice of which root one calls $\alpha$ (since the final formulas are symmetric).

For the quadratic case: $X=x{x}^{\prime }-by{y}^{\prime }$, $Y=x{y}^{\prime }+y{x}^{\prime }+ay{y}^{\prime }$.

For the cubic case (with the $(x,y,z)$-form):

• $X=x{x}^{\prime }-c(y{z}^{\prime }+z{y}^{\prime })+acz{z}^{\prime }$
• $Y=x{y}^{\prime }+y{x}^{\prime }-b(y{z}^{\prime }+z{y}^{\prime })-(ab-c)z{z}^{\prime }$
• $Z=x{z}^{\prime }+z{x}^{\prime }+y{y}^{\prime }+a(y{z}^{\prime }+z{y}^{\prime })+(a^2-b)z{z}^{\prime }$

Powers via binomial expansion

Setting ${\mathbf{x}}^{\prime }=\mathbf{x}$ in the composition law gives a squaring formula. For arbitrary $m$-th power, Lagrange writes $X_0+\alpha X_1+{\alpha }^2{X}_2+\cdots =(x_0+\alpha x_1+{\alpha }^2{x}_2+\cdots {)}^m,$ expands by Newton’s binomial theorem, and reduces ${\alpha }^k$ via the recurrence ${\alpha }^{k+n}=a_1{\alpha }^{k+n-1}-a_2{\alpha }^{k+n-2}+\cdots$. Comparing coefficients of $1,\alpha ,{\alpha }^2,\dots$ gives explicit polynomial formulas for $X_0,X_1,\dots$ in $x_0,x_1,\dots$ that satisfy $N_f(\mathbf{X})=N_f(\mathbf{x}{)}^m$.

Use in indeterminate analysis

To make a norm form $N_f(\mathbf{X})$ equal to an $m$-th power $V^m$, parametrise $\mathbf{X}$ by the power formula above; then $V=N_f(\mathbf{x})$ for any rational $\mathbf{x}$. In the cubic case this gives a free three-parameter family for $X^3+a{X}^{2}Y+\cdots =V^2$ (additions-9, Art. 92).

Lagrange uses this in the cubic case to solve the four-term equation $X^3+a{X}^{2}Y+bX{Y}^2+c{Y}^3=V^2$ by killing the $Z$-component via a single auxiliary equation in $x,y,z$.

Connection to later theory

This construction anticipates (source: additions-9, Art. 88 commentary):

• Gauss’s composition of binary quadratic forms (1801, Disquisitiones Arithmeticae §234ff): the quadratic case with $a=0$ gives the principal form $x^2+b{y}^2$; Gauss generalised to arbitrary discriminant.
• The norm of an algebraic number field: $N_f$ is the field norm $N_{K/\mathbb{Q}}$ on $K=\mathbb{Q}(\alpha )$.
• Class field theory and the connection between norms and units: solutions $N_f(\mathbf{x})=1$ correspond to units in $\mathbb{Z}[\alpha ]$ when $\alpha$ is an algebraic integer.

Related pages

• composition-of-forms
• binary-quadratic-forms
• brahmagupta-fibonacci-identity
• add9-norm-forms-composition
• ch2.0.11-quadratic-form-factorization
• ch2.0.12-quadratic-form-as-power
• vieta-formulas
• binomial-theorem