# What was the significance of Eisenstein's discovery of invariants?

I am trying to decipher a portion of James Joseph Sylvester's 1869 address entitled "The Study That Knows Nothing of Observation", which, among other things, surveys the landscape of 19th century mathematics.

In response to the self-posed question, "how did algebraic forms originate?", he replies,

"[they originated] in the accidental observation by Eisenstein, some score or more years ago, of a single invariant (the Quadrivariant of a Binary Quartic) which he met with in the course of certain researches ..."

I can't make sense of this, because it seems many of these mathematical objects have since been renamed. What are 'algebraic forms'? What are 'quadrivariants'? Who was Eisenstein, and what was the significance of his discovery?

The terminology did not change that much. Algebraic forms are more often called homogeneous polynomials, but for polynomials in two variables "binary form" is still often used. Invariant of a binary form is a polynomial in its coefficients that remains unchanged under the transformation of said coefficients induced by the action of the special linear group on form's variables. There is also a more general notion of covariant. The discriminant of a form is an invariant, and the determinnat of its Hessian is a covariant

Invariants of a binary form form a ring, and the study of invariants was a flourishing industry in 19-th century that Eisenstein helped to initiate, see Invariant of a binary form. Ferdinand Gotthold Max Eisenstein (1823-1852) was a brilliant German mathematician, who became fascinated by number theory after buying a French translation of Gauss's Disquisitiones Arithmeticae when he was 19. He was picked out as a young genius by Crelle of the famous journal in 1844, and became a lifelong protege of Alexander von Humboldt. Despite staying on the margins of 1848 revolution he was arrested with much detriment to his already poor health. Like Galois he died young, only 4 years later. He is perhaps best known for the Eisenstein criterion of irreducibility and the Eisenstein series in the theory of modular forms.

The ring of invariants of a binary quartic is generated by invariants of degrees 2 and 3. This ring is naturally isomorphic to the ring of modular forms of level 1, and the two generators correspond to the Eisenstein series $E_4$ and $E_6$ ("quadrivariant" must be the couterpart of $E_4$). Sylvester himself developed the theory of invariants and covariants of a binary quintic, and contributed to the theories of binary septic, octavic and duodecimic. But the "king of invariants" was Gordan. His signature achievement was to construct finite basis for invariants of binary forms of fixed order in 1868. He tried to extend the result to ternary and higher forms, but to no avail.

It all came crushing down after Hilbert proved the finite basis theorem in 1888. Except he "cheated", at least in the eyes of classical invariant theorists. Instead of constructing the basis Hilbert gave a pure existence proof based on ascending chain arguments. He submitted to Mathematische Annalen and Klein sent his paper to "king" Gordan, who opposed it and allegedly quipped "This is not mathematics, this is theology". At least, this is the traditional story that McLarty calls the "origin myth", propounded by Hilbert himself and others 30 years later, see McLarty's Theology and Its Discontents The Origin Myth of Modern Mathematics. In truth, Gordan and Hilbert cooperated on the 1888 paper, and Gordan's objection was to its "unfinished" state, not the non-constructive proof, and Hilbert "finished" the theory in 1893 by adding Nullstellensatz, etc.

"Gordan and most of his contemporaries were far too quick with their reasoning to notice the difference between a statement and its double negation. Without that distinction you cannot sharply distinguish proof by contradiction from direct proof... Apparently Gordan meant just what he said: the proof was not clear to him. When he published his own version he added that Hilbert’s ideas offered more help with calculating specific systems than Hilbert had bothered to use: "The proof Hilbert has given is entirely correct in substance; yet I feel a gap in his explication as he is satisfied to prove the existence of [solutions] without discussing their properties. To repair this gap I give a somewhat different proof..."

Klein ended up accepting Hilbert's 1888 paper without changes. In 1893 Gordan effectively developed what is now called Gröbner bases to construct invariants along Hilbert's lines, Emmy Noether merged both Hilbert's axiomatic method and Gordan's symbolic method in her work. But be it as it may, the effect of Hilbert's work was the rise of abstract algebra and the decline of classical invariant theory. For a good part of 20-th century it was almost forgotten until plucked out of obscurity by Rota in 1960-s, see The work of Gian-Carlo Rota on invariant theory by Grosshans. Buchberger introduced "Gröbner bases" in 1965, and with the rise of computers computational/algorithmic invariant theory made something of a comeback. Mumford's geometric invariant theory recovers iinvariants in a more abstract guise derived from Noether.

I recommend that you read some history of mathematics if you are interested in 19th century mathematics. A good one is Klein, Lectures on development of mathematics in 19th century, Chapter IV, section Theory of Invariants.

Eisenstein is Gotthold Eisenstein, a famous German mathematician, you may look in Wikipedia for his biography.

To clarify some thing implicit in comments and answers, from a more contemporary viewpoint:

Given a polynomial $$P(x_1,\ldots,x_n)$$ in $$n$$ variables, we can certainly do a linear change of variables by replacing $$x=(x_1,\ldots,x_n)$$ by $$y=(y_1,\ldots,y_n)=x\cdot A$$ for an $$n$$-by-$$n$$ matrix $$A$$. (And there are potentially issues about transpose, column-versus-row matrices, etc., but these are secondary.)

Reasonably-enough, in terms of colloquial English, a polynomial $$P$$ is invariant (for some subgroup $$G$$ of the group $$GL_n(\mathbb C)$$ of invertible $$n$$-by-$$n$$ complex matrices) when $$P\circ A=P$$ for all $$A$$.

The phenomenon whose 19-th and early-20-th century description is considerably at variance from our contemporary description is about "representations of the group (acting on the variables, hence, on the polynomials". In our contemporary terms, there are further, but/and highly structured, collections of polynomials beyond the simplest case of "invariant". Long ago, these had other names... I myself cannot reliably translate Sylvester's (quite creative! :) terminology into contemporary, for example.

But, in any case, it's about the representation theory of general linear groups, and subgroups theoreof, on spaces of polynomials. This has the effect of multi-linear algebra operations on the dual spaces of the representations of those groups on the vector spaces of "variables" themselves.