Fixed field

In algebra, the fixed-point subring ${\displaystyle R^{f}}$ of an automorphism f of a ring R is the subring of the fixed points of f:

More generally, if G is a group acting on R, then the subring of R:

is called the fixed subring or, more traditionally, the ring of invariants. In Galois theory, when R is a field and G is a group of field automorphisms, the fixed ring is a subfield called the fixed field of the automorphism group; see Fundamental theorem of Galois theory.

Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate rings of (affine or projective) GIT quotients and they play fundamental roles in the constructions in geometric invariant theory.

Example: Let ${\displaystyle R=k[x_{1},\dots ,x_{n}]}$ be a polynomial ring in n variables. The symmetric group S_n acts on R by permutating the variables. Then the ring of invariants R^G is the ring of symmetric polynomials. If a reductive algebraic group G acts on R, then the fundamental theorem of invariant theory describes the generators of R^G.

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