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Results tagged with ra.rings-and-algebras
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user 15505
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
7
votes
1
answer
795
views
Factorization of an irreducible polynomial in the field extension it defines
In field theory, the following fact is used in the construction of splitting fields: Given a field $F$ and an irreducible polynomial $f \in F[x]$, the quotient $F[\alpha]/(f(\alpha))$ is a field exten …
- 2,017
10
votes
If two monic polynomials of $\mathbb{Z}_p[X]$ (p-adic integer coefficient) are relatively pr...
If the two polynomials are $f,g$, then the $\mathbb{Z}_{p}$-module $M := \mathbb{Z}_{p}[X]/(f,g)$ is finitely generated (since at least one of $f,g$ is monic) and satisfies $M \otimes_{\mathbb{Z}_{p}} …
- 2,017