Definition:Irreducible
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Irreducible or irreducibility may refer to:
- Ring Theory:
- Irreducible Element: An element of an integral domain $D$ is irreducible if and only if it has no non-trivial factorization in $D$.
- Irreducible Polynomial: A polynomial is irreducible over a field if and only if it cannot be represented as the product of two or more non-constant polynomials over that field.
- Topology:
- Irreducible Space: A topological space is irreducible if and only if it cannot be represented as a decomposition of closed sets.
- Irreducible Subset: a subset of a topological space which is irreducible
- Representation Theory:
- Irreducible Linear Representation: A linear representation $\rho: G \to \GL V$ is irreducible if and only if it has no non-trivial subspace $W$ which is invariant for every linear operator in the set $\set {\map \rho g: g \in G}$.
- Irreducible $G$-Module: A $G$-module is irreducible if and only if its corresponding linear representation is irreducible.
- Irreducible Representation (Lie Algebra)
- Irreducible Representation (Topological Group)