How do the definitions of irreducible and prime elements differ? The implication "irreducible implies prime" is true in integral domains in which any two non-zero elements have a greatest common divisor This is for instance the case of unique factorization domains
What is the meaning of an irreducible representation? @okj: An irreducible representation is a map from the group to a group of matrices; under the representation (under the map), each element of the group will map to a matrix You can think of an irreducible representation as a way to assign to every element of the group (in this case, SO (3)), a particular matrix (linear transformation)
Irreducible polynomial means no roots? - Mathematics Stack Exchange The condition of being irreducible if it doesn't have any roots is false Consider, for example, the polynomial $$ x^4 + 4 x^2 + 3 = (x^2 + 1) (x^2 + 3) \in \mathbb {R} [x] $$ When the coefficient ring is not a field, though, some coefficients are not invertible The polynomial $$ 2x \in \mathbb {Z} [x]$$
abstract algebra - What is meant by a polynomial that is irreducible . . . So, when you talk about a polynomial being irreducible or prime, one runs into the same sort of reason that $1$ is not prime, because the notion of irreducibility is only a useful concept when you ignore the constants since you can always multiple or divide by a constant and nothing really changes much