# 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".

135 questions
3answers
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### Proving $f(x)=x^4+4x^3+3x^2+7x-4$ is irreducible

Here's my attempt, I'm almost there but I'm stuck: Using a hint, I wrote the modular reduction: Reducing the coefficients modulo $2$ gives: $\left [ f \right ]_2=x^4+x^2+x=x(x^3+x+1)$. Reducing ...
1answer
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### Polynomial rings and ideals

I'm trying to learn some algebra by myself and I need some help. $I=(x^2,x^3)$ an ideal in $R[X]$. Give an example of two polynomials with exactly four terms, one that is in $I$ and one that isn't. ...
2answers
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### Subrings $A$ of $\mathbb{F}_p[x]$ such that $\dim_{\mathbb{F}_p}\mathbb{F}_p[x]/A=1$.

Suppose we have the ring $\mathbb{F}_p[x]$ of polynomials with coefficients in the Galois field with $p$ prime number elements. I want to find all subrings $A$ containing the identity such that when ...
1answer
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2answers
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0answers
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### Surjections of polynomial rings, Krull dimension, and regular sequences

Let $k$ be a commutative Noetherian ring with unity (I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a surjective $k$-algebra homomorphism....
0answers
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### Hilbert polynomials in two variables with Macaulay2

In J. Symb. Comput. (1999) 28, 681-710, Levin worked with bifiltered, finitely generated $R$-modules ($R$ being a polynomial ring in two sets of variables) and he found an analogue of the Hilbert ...
1answer
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### Describe explicitly $\text{spec}(\mathbb{R}[x])$ and $\text{spec}(\mathbb{C}[x])$ [closed]

Describe explicitly $\text{spec}(\mathbb{R}[x])$ and $\text{spec}(\mathbb{C}[x])$, (where for a given ring $R$, $\text{spec}(R)$ is defined to be the set of all prime ideals of $R$). I don't have an ...
2answers
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### Zero divisors of $\mathbb{Z}_{7}[x] / (x^4+x^3-3)$ and inverse element of $\overline{x+1}$

How many zero divisors there are in the ring $\mathbb{Z}_{7}[X] / (x^4+x^3-3)$? What is the inverse element of $\overline{x+1}$? I'm not sure where to begin, so I thought it might be a good idea ...
2answers
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1answer
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### Does there always exist a linear shift of a given polynomial such that all coefficients are nonzero?

Let $K$ be a field and let $f(x)=a_0+a_1x+\cdots +a_nx^n\in K[x]$ be such that $a_n\neq 0$. Does there always exists some $\alpha\in K$ such that the coefficients of $f(x-\alpha)$ are nonzero in every ...
1answer
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1answer
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### Applying degree 2 or 3 irreducibility tests to higher degree

Given the polynomial$\ x^4+x+1$, I have to find out if it is irreducible over $\mathbb Q$. When looking at the solutions, they applied the degree 2 or 3 irreducibly tests to determine that it ...
2answers
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### Find all of homomorphisms $Φ$ from $\mathbb C$[$x$]/$I$ to $\mathbb C$ that satisfies specific condition

I want to know how can I find ALL homomorphism that satisfies the condition mentioned in the question above. Please give me a method or answer in detail.
2answers
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### GCDs for the polynomial ring over a Galois field.

You can find many examples of computing the inverse of an element inside a Galois field. (For example here) What happens if we look at the polynomial ring over a Galois field and would like to ...
2answers
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0answers
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### Field of fractions and rational functions

When I have a field $k$ and take the ring of polynomials in the variables $x1, x2, ..., xn$, and subsequently take the quotient field of these polynomials, I was asking myself, is this in 1:1 ...
2answers
206 views

### Finding units and zero divisors in a polynomial quotient ring

I am trying to study for an exam and I am not sure of this solution my professor posted to an exercise. I am given the polynomial quotient ring $\mathbb{Z}_6/(x^2+2x)$ and have to find all units and ...
2answers
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### Isomorphism between quotient fields of polynomial rings [closed]

Let $R$ be an integral domain and $F$ be its field of fractions. If $X$ is a nonempty set, prove that there is a ring monomorphism from $R[X]$ to $F[X]$ that extends to an isomorphism of their ...
1answer
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### Homomorphisms and automorphisms on polynomial rings

I am trying to prove a series of propositions: Given any homomorphism p from $\mathbb{R}$[X] to $\mathbb{R}$[X], show that it is equal to $\phi_g$ for a unique g in $\mathbb{R}$[X], with $\phi_g$(f) =...
0answers
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### Classifying homomorphisms on polynomial rings with real coefficients.

Show that every homomorphism $\mathbb{R}$[X] $\rightarrow$ $\mathbb{R}$[X] can is equal to $φ_g$ for a unique g $\in$ $\mathbb{R}$[X], given by $φ_g(f)$ = $f(g(X))$ My guess for any homomorphism $h$, ...
0answers
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### Answer verification to question about polynomial rings and derivatives.

Can someone check whether my answers are okay to the following Judson question? (a) Let $f(x)=a_{n}x^{n}+\cdots +a_{m}x^{m}+\cdots a_{1}x+a_{0}$ and $g(x)=b_{m}x^{m}+\cdots +b_{1}x+b_{0}$. Assume ...
1answer
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1answer
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### If $f|d$, then $f$ is invertible in $R[X]/(d)R[X]$

Let $R$ be a field and $f$ and $d$ be polynomials in $R[X]$. If $f|d$, then $f$ is invertible in $R[X]/(d)R[X]$. I tried to either prove of disprove this statement, but so far I haven't been able ...
1answer
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### No element of degree $0$ (constant $= a_0 X^0$) in $\mathbb{Z}_4/(2X) \mathbb{Z}_4$

Consider $\mathbb{Z}_4[X]/(2X) \mathbb{Z}_4[X]$. Then we wish to show/verify that the residue class $X$ does not contain an element of degree $0$. In the previous exercise the book asked that if we ...
2answers
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### $a^4=0$ then $1-a$ is invertible in $R[x]/(d)R[x]$ [duplicate]

Suppose $a^4=0$, for some $a \in R[x]/(d)R[x]$, then prove that $1-a$ is invertible. I was thinking since $a^4 = a \cdot a \cdot a \cdot a=0$, this implies that $a$ has to be zero (?) . Now we have ...
1answer
51 views

### The maps $f_+$ and $f_-$ are well-defined.

We define the maps $f_+$ and $f_-$ from $\mathbb{Q}[X]/(X^2 -2) \mathbb{Q}[X]$ to $\mathbb{Q} + \mathbb{Q} \cdot \sqrt2$ in the following manner: For any residue class $g+ (X^2-2) \mathbb{Q}[X]$, we ...