- 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
- abstract algebra - Methods to see if a polynomial is irreducible . . .
Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now First is Eisenstein criterion Another is that if a polynomial is irreducible mod p th
- Proving that a polynomial is irreducible over a field
Proving that a polynomial is irreducible over a field Ask Question Asked 14 years, 1 month ago Modified 4 years, 8 months ago
- 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)
- abstract algebra - Why an absolutely irreducible representation is . . .
Why an absolutely irreducible representation is irreducible under all field extensions? Ask Question Asked 9 years, 6 months ago Modified 4 years, 5 months ago
- 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 $$ 2x \in \mathbb {Z} [x]$$
- What is an irreducible matrix? - Mathematics Stack Exchange
What is an irreducible matrix? Ask Question Asked 6 years, 7 months ago Modified 6 years, 7 months ago
- linear algebra - Irreducible vs. indecomposable representation . . .
However, Serre is dealing with finite-dimensional complex representations of finite groups, and in that case, yes, every indecomposable representation is irreducible
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