Composition of Forms

Summary: A bilinear law $\mathbf{X}=\mathbf{x}\star {\mathbf{x}}^{\prime }$ on the indeterminates of a norm form $N_f$ such that $N_f(\mathbf{x})\cdot N_f({\mathbf{x}}^{\prime })=N_f(\mathbf{X})$. For the quadratic form $x^2+axy+b{y}^2$ this is the brahmagupta-fibonacci-identity; for higher-degree norm forms Lagrange gives the explicit formulas in terms of the coefficients of the underlying polynomial.

Sources: additions-9 (Articles 89, 91), additions-7 (Article 75 for the Pell composition).

Last updated: 2026-05-10.

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Quadratic case

For the form $\Phi (x,y)=x^2+axy+b{y}^2$ associated to $s^2-as+b=0$: $\Phi (x,y)\cdot \Phi (x^{\prime },y^{\prime })=\Phi (X,Y),X=x{x}^{\prime }-by{y}^{\prime },Y=x{y}^{\prime }+y{x}^{\prime }+ay{y}^{\prime }.$ With $a=0$ this is the classical brahmagupta-fibonacci-identity $(x^2+b{y}^2)(x^{\prime 2}+b{y}^{\prime 2})=(x{x}^{\prime }-by{y}^{\prime }{)}^2+b(x{y}^{\prime }+y{x}^{\prime }{)}^2.$ Iterating the law gives the squaring formula $\Phi (x,y{)}^2=\Phi (x^2-b{y}^2,2xy+a{y}^2)$ and, for general $m$, formulas via the recurrence ${\alpha }^k=A^{(k)}\alpha -B^{(k)}$ with $A^{(k+1)}=a{A}^{(k)}-b{A}^{(k-1)}$ (additions-9, Art. 90).

Pell composition

The quadratic composition specialises in two important ways:

Pell case ($a=0$, $b=-A$): for $\Phi (p,q)=p^2-A{q}^2$, $(p^2-A{q}^2)(p^{\prime 2}-A{q}^{\prime 2})=(p{p}^{\prime }+Aq{q}^{\prime }{)}^2-A(p{q}^{\prime }+p^{\prime }q{)}^2.$ This is the engine behind the Pell composition $t\pm u\sqrt{A}=(T\pm V\sqrt{A}{)}^m$ of additions-7, Art. 75: each multiplication of a unit-norm element by $T+V\sqrt{A}$ produces another unit-norm element.

Cyclotomic case ($a=-1$, $b=1$): for $\Phi (x,y)=x^2-xy+y^2$ (norm form of $\mathbb{Z}[{\zeta }_6]$).

Cubic case

For $\Phi (x,y,z)$ associated to $s^3-a{s}^2+bs-c=0$ (the ten-term form of add9-norm-forms-composition): $\Phi (x,y,z)\cdot \Phi (x^{\prime },y^{\prime },z^{\prime })=\Phi (X,Y,Z)$ with

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

The composition is associative and commutative, since it descends from polynomial multiplication in $\mathbb{Z}[\alpha ,\beta ,\gamma ]/⟨s^3-a{s}^2+bs-c⟩$ followed by symmetrisation over the Galois action permuting $\alpha ,\beta ,\gamma$.

Higher degrees

The Article 94 sketch indicates the same construction works for any degree $n$: compose by polynomial multiplication mod the defining equation. In modern terms, this is multiplication in the ring $\mathbb{Z}[s]/(f(s))$, and the norm-multiplicativity is the determinant-multiplicativity of multiplication-by-$\xi$ as a $\mathbb{Z}$-linear endomorphism of $\mathbb{Z}[s]/(f(s))$.

Use as solution machine

Composition laws give a one-shot procedure to convert one solution of $N_f(\mathbf{x})=N$ into infinitely many: compose with any element of norm $1$ (a “unit”). This is the principle behind:

• The pell-equation solution generator: from $(t_1,u_1)$ with $t_1^2-A{u}_1^2=1$, all $(t_m,u_m)$ are obtained by composition.
• The infinite families of ch2.0.6-integer-solutions-quadratic-squares: seed solution composed with Pell solutions.
• Lagrange’s parametric solution of $X^3+a{X}^{2}Y+bX{Y}^2+c{Y}^3=V^2$ (additions-9, Art. 92).

Related pages

• norm-forms
• brahmagupta-fibonacci-identity
• binary-quadratic-forms
• pell-equation
• add9-norm-forms-composition
• add7-integer-quadratic-method
• ch2.0.11-quadratic-form-factorization
• ch2.0.12-quadratic-form-as-power