First and second fundamental theorems of invariant theory

In algebra, the first and second fundamental theorems of invariant theory concern the generators and the relations of the ring of invariants in the ring of polynomial functions for classical groups (roughly the first concerns the generators and the second the relations). The theorems are among the most important results of invariant theory.
Classically the theorems are proved over the complex numbers. But characteristic-free invariant theory extends the theorems to a field of arbitrary characteristic.
First fundamental theorem.
The theorem states that the ring of formula_1-invariant polynomial functions on formula_2 is generated by the functions formula_3, where formula_4 are in formula_5 and formula_6.
Second fundamental theorem for general linear group.
Let "V", "W" be finite dimensional vector spaces over the complex numbers. Then the only formula_7-invariant prime ideals in formula_8 are the determinant ideal
formula_9
generated by the determinants of all the formula_10-minors.