How do the definitions of "irreducible" and "prime" elements differ?
The most widespread terminology would be “irreducible” for that case. However there's no international math police, so everybody is entitled the right of naming concepts as they like; …
abstract algebra - Methods to see if a polynomial is irreducible ...
Oct 9, 2021 · Osada's criterion. Let f(x) = + + ⋯ + x ± p be a polynomial with integer coefficients, where p is a prime. If p> 1 + + ⋯ +, then f(x) is irreducible over Z. Following one is also simple …
What is an irreducible matrix? - Mathematics Stack Exchange
In general, a matrix is reducible if and only if it is permutation-similar to a block-triangular matrix of the form (X 0 Y Z) (X Y 0 Z), where X X and Z Z are square sub-blocks (of possibly different …
What is the meaning of an "irreducible representation"?
May 13, 2011 · 2. @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 …
commutative algebra - Prime ideals and irreducible ideals
1. In any commutative ring with unity, prime ideals are proper (by definition), irreducible, and radical. As noted in comments and answers above, the converse is true: proper, irreducible, …
How can I prove that something is an irreducible element?
Here's a different approach. Since prime elements are irreducible, it suffices to show that $4+i$ is prime. (In fact, since $\mathbb{Z}[i]$ is a UFD, then an element is irreducible iff it is prim
What are irreducible factors? - Mathematics Stack Exchange
In brief, a polynomial p is irreducible if p = qr implies one of the "factors" q, r is a unit (invertible ring element). The definition applies to rings and especially to integral domains. The notion of …
What are examples of irreducible but not prime elements?
Then $\,x\,$ is irreducible but not prime, since $\,x\mid (\sqrt 2 x)^2\,$ but $\,x\nmid \sqrt 2 x,\,$ by $\sqrt 2\not\in \Bbb Q$. It should be clear how to give many further examples of this sort. …
abstract algebra - Is the ideal generated by irreducible element in ...
Suppose r is irreducible in a PID say D. Let I be an ideal of D containing (r) the ideal generated by r. Since D is a principal ideal domain, there exist s in D such that I=(s), therefore (r) is a subset …
How do I find out if a polynomial is irreducible?
Dec 4, 2014 at 5:57. @khajvah: as Robert Israel notes, your polynomial is reducible. But it is a general technique for showing a polynomial is irreducible over Q Q. Note that you need not …