add9-norm-forms-composition

Additions Chapter IX — Algebraic Forms Producing Similar Functions

Summary: Lagrange shows that for each algebraic equation $s^n-a{s}^{n-1}+\cdots =0$ with roots $\alpha ,\beta ,\dots$, the symmetric norm form ${\prod }_i(x+{\alpha }_{i}y+{\alpha }_i^{2}z+\cdots )$ is multiplicatively closed: products and powers of two such forms are again of the same shape. This anticipates Gauss’s composition of binary quadratic forms and the modern theory of norm forms of number fields.

Sources: additions-9 (Articles 88–94).

Last updated: 2026-05-10.

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This chapter is the Appendix to Chapters XI and XII of Euler’s Part II. Lagrange uses irrational and even imaginary algebraic factors to construct, for each degree $n$, a homogeneous polynomial form in $n$ variables that behaves multiplicatively — providing a master tool for problems “make this form a square / cube / $m$-th power.”

Historical preamble (Art. 88)

Lagrange notes that, around 1768, he and Euler independently arrived at the idea of using irrational and imaginary factors to study Diophantine forms (cf. ch2.0.11-quadratic-form-factorization and ch2.0.12-quadratic-form-as-power). He presented his version to the Academy in a Memoir summarised at the end of his 1767 Recherches sur les Problèmes Indéterminés (printed 1769, before the German edition of Euler’s Algebra). This chapter generalises the method to arbitrary degree.

The quadratic form (Art. 89)

Let $\alpha ,\beta$ be roots of $s^2-as+b=0$, so $\alpha +\beta =a$ and $\alpha \beta =b$. Then $(x+\alpha y)(x+\beta y)=x^2+axy+b{y}^2.$ For two such forms with parameters $(x,y)$ and $(x^{\prime },y^{\prime })$: $[(x+\alpha y)(x^{\prime }+\alpha y^{\prime })]\cdot [(x+\beta y)(x^{\prime }+\beta y^{\prime })]=X^2+aXY+b{Y}^2,$ where, using ${\alpha }^2=a\alpha -b$ to eliminate ${\alpha }^2$ from the partial product, $\boxed{X=x{x}^{\prime }-by{y}^{\prime },Y=x{y}^{\prime }+y{x}^{\prime }+ay{y}^{\prime }.}$ The product of any number of such forms is again of the same shape — this is the composition law. With $a=0$ it specialises to the brahmagupta-fibonacci-identity $(x^2+b{y}^2)(x^{\prime 2}+b{y}^{\prime 2})=(x{x}^{\prime }-by{y}^{\prime }{)}^2+b(x{y}^{\prime }+y{x}^{\prime }{)}^2$.

Powers of the quadratic form (Art. 90)

Setting $x^{\prime }=x,y^{\prime }=y$ gives the squaring law: $(x^2+axy+b{y}^2{)}^2=X^2+aXY+b{Y}^2,X=x^2-b{y}^2,Y=2xy+a{y}^2.$ For the cube, iterating once more: $X^{\prime }=x^3-3bx{y}^2+ab{y}^3,Y^{\prime }=3x^{2}y+3ax{y}^2+(a^2-b)y^3.$

For a general $m$-th power, Lagrange writes $X+\alpha Y=(x+\alpha y{)}^m$ and expands by Newton’s binomial theorem. Using ${\alpha }^2=a\alpha -b$ recursively gives ${\alpha }^k=A^{(k)}\alpha -B^{(k)}$ with the linear recurrence $A^{(1)}=1,B^{(1)}=0,A^{(2)}=a,B^{(2)}=b,A^{(k+1)}=a{A}^{(k)}-b{A}^{(k-1)},B^{(k+1)}=a{B}^{(k)}-b{B}^{(k-1)}.$ Comparing rational and $\alpha$-parts of the binomial expansion gives: $X=x^m-\left(\genfrac{}{}{0ex}{}{m}{1}\right)x^{m-1}y{B}^{(1)}-\left(\genfrac{}{}{0ex}{}{m}{2}\right)x^{m-2}{y}^2{B}^{(2)}-\left(\genfrac{}{}{0ex}{}{m}{3}\right)x^{m-3}{y}^3{B}^{(3)}-\cdots$ $Y=\left(\genfrac{}{}{0ex}{}{m}{1}\right)x^{m-1}y{A}^{(1)}+\left(\genfrac{}{}{0ex}{}{m}{2}\right)x^{m-2}{y}^2{A}^{(2)}+\left(\genfrac{}{}{0ex}{}{m}{3}\right)x^{m-3}{y}^3{A}^{(3)}+\cdots$ and these satisfy $X^2+aXY+b{Y}^2=(x^2+axy+b{y}^2{)}^m$.

When $a=0$ the recursion has period 4 ($A^{(k)}=1,0,-b,0,b^2,0,-b^3,\dots$; $B^{(k)}=0,b,0,-b^2,0,b^3,\dots$), giving the cleaner closed forms used in ch2.0.12-quadratic-form-as-power: $X=x^m-\left(\genfrac{}{}{0ex}{}{m}{2}\right)x^{m-2}{y}^{2}b+\left(\genfrac{}{}{0ex}{}{m}{4}\right)x^{m-4}{y}^4{b}^2-\cdots$ $Y=m{x}^{m-1}y-\left(\genfrac{}{}{0ex}{}{m}{3}\right)x^{m-3}{y}^{3}b+\left(\genfrac{}{}{0ex}{}{m}{5}\right)x^{m-5}{y}^5{b}^2-\cdots$ with $X^2+b{Y}^2=(x^2+b{y}^2{)}^m$.

The cubic form (Art. 91)

Let $\alpha ,\beta ,\gamma$ be roots of $s^3-a{s}^2+bs-c=0$, so $\alpha +\beta +\gamma =a$, $\alpha \beta +\alpha \gamma +\beta \gamma =b$, $\alpha \beta \gamma =c$. Compute the symmetric product $(x+\alpha y+{\alpha }^{2}z)(x+\beta y+{\beta }^{2}z)(x+\gamma y+{\gamma }^{2}z).$ Using the identities ${\alpha }^2+{\beta }^2+{\gamma }^2=a^2-2b$, ${\sum }_{i\ne j}{\alpha }_i^2{\alpha }_j=ab-3c$, $\sum {\alpha }_i^2{\alpha }_j^2=b^2-2ac$, $\sum {\alpha }_i^2{\alpha }_j{\alpha }_k=ac$, $\sum {\alpha }_i^2{\alpha }_j^2{\alpha }_k=bc$, the product evaluates to the ten-term cubic norm form $\boxed{N(x,y,z)=x^3+a{x}^{2}y+(a^2-2b)x^{2}z+bx{y}^2+(ab-3c)xyz+(b^2-2ac)x{z}^2+c{y}^3+ac{y}^{2}z+bcy{z}^2+c^2{z}^3.}$ The composition of two such forms with parameters $(x,y,z)$ and $(x^{\prime },y^{\prime },z^{\prime })$ produces a third with $X=x{x}^{\prime }-c(y{z}^{\prime }+z{y}^{\prime })+acz{z}^{\prime }$ $Y=x{y}^{\prime }+y{x}^{\prime }-b(y{z}^{\prime }+z{y}^{\prime })-(ab-c)z{z}^{\prime }$ $Z=x{z}^{\prime }+z{x}^{\prime }+y{y}^{\prime }+a(y{z}^{\prime }+z{y}^{\prime })+(a^2-b)z{z}^{\prime }$ satisfying $N(X,Y,Z)=N(x,y,z)\cdot N(x^{\prime },y^{\prime },z^{\prime })$.

Squares and powers of the cubic form (Arts. 92–93)

Squaring (Art. 92). Setting $(x^{\prime },y^{\prime },z^{\prime })=(x,y,z)$: $X=x^2-2cyz+ac{z}^2,Y=2xy-2byz-(ab-c)z^2,Z=2xz+y^2+2ayz+(a^2-b)z^2$ satisfy $N(X,Y,Z)=V^2$ with $V=N(x,y,z)$.

Application: solving $X^3+a{X}^{2}Y+bX{Y}^2+c{Y}^3=V^2$ (the “shortened” form, just the four-term subsystem). Choose $z$ to kill $Z$: set $2xz+y^2+2ayz+(a^2-b)z^2=0$, giving $x=-\frac{y^2+2ayz+(a^2-b)z^2}{2z}.$ Substituting back yields rational two-parameter solutions of the cubic-form-square equation — Lagrange notes “this solution deserves particular attention, on account of its generality, and the manner in which we have arrived at it; which is, perhaps, the only way in which it can be easily resolved.”

General $m$-th power (Art. 93). Set $X+\alpha Y+{\alpha }^{2}Z=(x+\alpha y+{\alpha }^{2}z{)}^m$ and expand. Defining $(x+\alpha y+{\alpha }^{2}z{)}^m=P+P^{\prime }\alpha +P^{\prime \prime }{\alpha }^2+P^{\prime \prime \prime }{\alpha }^3+\cdots$, the $P^{(k)}$ satisfy the recurrences $P=x^m,P^{\prime }=\frac{myP}{x},P^{\prime \prime }=\frac{(m-1)y{P}^{\prime }+2mzP}{2x},P^{\prime \prime \prime }=\frac{(m-2)y{P}^{\prime \prime }+(2m-1)z{P}^{\prime }}{3x},\dots$ (“easily demonstrated by the differential calculus”). Reducing ${\alpha }^k=A^{(k)}{\alpha }^2-B^{(k)}\alpha +C^{(k)}$ via ${\alpha }^3=a{\alpha }^2-b\alpha +c$ with the three-term recurrence $A^{(k+1)}=a{A}^{(k)}-b{A}^{(k-1)}+c{A}^{(k-2)}$ (and likewise for $B^{(k)},C^{(k)}$), one extracts $X={\sum }_k{P}^{(k)}{C}^{(k)},Y=-{\sum }_k{P}^{(k)}{B}^{(k)},Z={\sum }_k{P}^{(k)}{A}^{(k)}$ satisfying $N(X,Y,Z)=N(x,y,z{)}^m$.

The quartic case (Art. 94, sketch)

For $s^4-a{s}^3+b{s}^2-cs+d=0$ with roots $\alpha ,\beta ,\gamma ,\delta$, the same construction uses the four factors $x+\alpha y+{\alpha }^{2}z+{\alpha }^{3}t,\text{etc.}$ Computing their symmetric product directly via Newton-Girard identities is tedious, so Lagrange proposes an elimination shortcut: setting $\rho =x+sy+s^{2}z+s^{3}t$ and eliminating $s$ from $\rho$ and the quartic, one obtains a quartic in $\rho$: ${\rho }^4-N{\rho }^3+P{\rho }^2-Q\rho +R=0$ whose four roots are the four factors. The constant term $R$ — the product of all roots — is therefore the desired norm form. Lagrange does not develop the explicit form, ending with: “we have now said enough on this subject, which we might resume, perhaps, on some other occasion.”

Significance

This chapter contains, avant la lettre:

• Gauss’s composition of binary quadratic forms (Art. 89, $a=0$ case) — explicit composition law for $x^2+b{y}^2$.
• The general theory of norm forms of an algebraic number field — Art. 91 is the norm form $N_{K/\mathbb{Q}}(x+\alpha y+{\alpha }^{2}z)$ for $K=\mathbb{Q}(\alpha )$ a cubic field.
• The connection between norm-multiplicativity and symmetric functions of conjugates — the fact that the $X,Y,Z$ formulas don’t depend on which root is called $\alpha$.

Lagrange’s closing line — “perhaps they have already been found too long” — masks one of the most consequential proto-algebraic developments of the eighteenth century.

Related pages

• norm-forms
• composition-of-forms
• binary-quadratic-forms
• brahmagupta-fibonacci-identity
• ch2.0.11-quadratic-form-factorization
• ch2.0.12-quadratic-form-as-power
• fermats-last-theorem-n3
• algebraic-identities
• binomial-theorem
• vieta-formulas