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Questions tagged [polynomial-rings]

This tag is used for questions on polynomials rings in an arbitrary number of variables related to different topics studied in ring theory and commutative algebra. Questions related to high-school polynomials level or similar should use the tag "polynomials".

4
votes
2answers
1k views

Irreducibility criteria for polynomials with several variables.

Let $K$ be a field. Show that $x^2-yz$ is irreducible in $K[x,y,z]$. Deduce that $x^2-yz$ is prime. If it is $K[x]$, then there are several methods which can be used to check whether a given ...
2
votes
2answers
238 views

Defining irreducible polynomials over polynomial rings

Let R be a ring, and R[x] be a polynomial ring. Can we define what it means for a polynomial $p(x) \in R[x]$ to be irreducible over R[x]? Various sources (such as Wikipedia) only provide such ...
28
votes
2answers
16k views

How to deal with polynomial quotient rings

The question is quite general and looks to explore the properties of quotient rings of the form $$\mathbb{Z}_{m}[X] / (f(x)) \quad \text{and} \quad \mathbb{R}[X]/(f(x))$$ where $m \in \mathbb{N}$ ...
2
votes
3answers
80 views

Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that the $\gcd$ of $x^5$ and $x^6$ does not exist

Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that a gcd of $x^5$ and $x^6$ does not exist in $R$. I am reasoning for the absurd and I am ...
2
votes
2answers
68 views

Trying to prove that $\langle y-x^2, z-x^3\rangle$ is prime in $K[x,y,z]$

Let $K$ be a field. I am trying to prove that $I=\langle y-x^2, z-x^3\rangle$ is a prime ideal in $K[x,y,z]$. Main idea: Define a homomorphism $\phi:K[z,y,z]\rightarrow K[x]$ as $$\phi(x)=x, \phi(y)...
1
vote
1answer
591 views

Finding all the ideals of $\mathbb{Z} [x]/(2, (x^3+1))$?

This is from Dummit and Foote (Section 9.2): 7. Determine all the ideals of the ring $\mathbb{Z} [x]/(2, (x^3+1))$. This is my attempt to understand what's going on: My plan is to find a nice ...
2
votes
1answer
85 views

Show that every nonzero prime ideal in $\mathbb{C}[x, y]/(x^2 - y^2 - 1)$ is maximal [duplicate]

Show that the ring $A := \mathbb{C}[x, y]/(x^2 - y^2 - 1)$ is an integral domain. Further show that every nonzero prime ideal in A is maximal. I proved that $A$ is an integral domain by showing that $...