# 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".

7 questions
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 ...
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 ...
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 $m \in \mathbb{N}$ ...
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 ...
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)...
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 ...
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 \$...