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m →Terminology: The invariants "are" not an algebra. They form an algebra (together). Alternatively, the set of invariants is a subalgebra of the polynomial ring over the variables a_0,..., a_n |
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In terms of [[representation theory]], given any representation ''V'' of the group ''SL''<sub>2</sub>('''C''') one can ask for the ring of invariant polynomials on ''V''. Invariants of a binary form of degree ''n'' correspond to taking ''V'' to be the (''n'' + 1)-dimensional irreducible representation, and covariants correspond to taking ''V'' to be the sum of the irreducible representations of dimensions 2 and ''n'' + 1.
The invariants of a binary form
Forms of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10 are sometimes called quadrics, cubic, quartics, quintics, sextics, septics or septimics, octics or octavics, nonics, and decics or decimics. "Quantic" is an old name for a form of arbitrary degree. Forms in 1, 2, 3, 4, ... variables are called unary, binary, ternary, quaternary, ... forms.
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