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I am reading an Italian encyclopedia on the history of Invariant Theory. This was an early branch of abstract algebra that was big in the middle of the nineteenth century.

Here is a line that I don't understand which I hope that some of you people might answer:

$u= ae- 4bd+3c^2$ is an invariant of $A=ax^4 + 4bx^{3}y + 6cx^{2}y^{2}+ 4dxy^{3}+ey^{4}$

Why is $u$ an invariant for this particular homogenous form?

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Are you looking for a proof of the statement or an explanation of what it is saying? It says $u$ is invariant under the "change of variables" action on the space of binary forms of degree $4$. The naive way to verify it is to perform a horrendously messy calculation to write the coefficients ($a',b',c',d',e'$) of the form obtained by change of variables in terms of the original coefficients and show that $a'e' - 4b'd' + 3(c')^2 = ae - 4bd + 3c^2$. –  Michael Joyce Mar 28 '12 at 14:22
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