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- 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 - Is the ideal generated by irreducible element in . . .
An element is irreducible iff the ideal it generates is maximal amongst the principal ideals If all ideals are principal, then an element is irreducible iff the ideal it generates is maximal
- 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 $$ 2x \in \mathbb {Z} [x]$$
- Irreducibles are prime in a UFD - Mathematics Stack Exchange
On the RHS, we know $a$ is irreducible If $d$ is invertible (i e an unit), then RHS only has $1$ irreducible element, which contradicts the definition of UFD
- general topology - Irreducible subsets of a topological space . . .
So could you provide some examples of irreducible subsets of a not-too-exotic topological space? A minor question: why is it made explicit the fact that the empty set is not irreducible?
- Ring of polynomials over a field has infinitely many primes
This discussion only gets you the irreducible polynomials of a fixed degree r Which in a finite field you couldn't possibly hope to get infinitely many of My question regards proving there are inf many irreducible polynomials over a fixed field but the degree of the polynomials can be arbitrarily large
- Prove that Spec $R$ is irreducible if and only if the radical ideal . . .
2 An easy way is to use the fact that irreducible components of the spectrum of a ring correspond to the minimal primes in the ring, see the Stacks Project
- 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
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