0
votes
0answers
12 views

Presentation of a module by generators and relations

Let $R:=\mathbb C[T]$. Match the $R$-module with the presentation by generators and relations. $\bullet$$R$-modules: $M:=\mathbb C[T,T^{-1}]$ (Laurent Polynomials)$\qquad$$N:=\mathbb ...
5
votes
3answers
45 views

Show some polynomial is irreducible over the field of 7 elements.

I have to show that the polynomial $x^4+x^3+x^2+x+1$ is irreducible over the field $F_7$. It doesn't have roots in $F_7$, but I can't show it does not have degree two irreducible factors in $F_7[x]$. ...
1
vote
1answer
47 views

If $k[X]/f = k[X]/g$, does $f = g$?

Let $k$ be a field and $f, g$ be irreducible monic polynomials in $k[X]$. Suppose $k[X]/f \stackrel{\sim}{=} k[X]/g$. Then does $f = g$? If so, how can this be generalized? Otherwise, how should I ...
0
votes
1answer
14 views

A question about co-prime polynomials in $\Bbb{C}[x,y]$

Say $f$ and $g$ are two co-prime polynomials in $\Bbb{C}[x,y]$. Can the following expression always be written $$af+bg=1$$ where $a,b,f,g\in\Bbb{C}[x,y]$? I realise that the Euclidean algorithm is not ...
0
votes
1answer
16 views

Show that there is no polynomial $f \in (\mathbb{Z}/100\mathbb{Z})$ satisfying f(1)=1 and f(11)=17

As stated in the title. This is part of a homework assignment.
0
votes
3answers
28 views

Quotient Rings of Rings in Several Variables

How should I interpret $$\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right)?$$ Is $$\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right) \cong \left(\left(\mathbb{Z[x,y]}/(y) \right)/(x) \right)?$$ ...
2
votes
1answer
37 views

What are the units in $R[x_1, x_2, … , x_n]$? Or $R[x]$?

I'm self teaching abstract algebra and I've come across this question...I cannot figure it out! R is not assumed to be a domain so it can't just be the constant polynomials, but whenever I try to ...
0
votes
0answers
57 views

Estimating the polynomial from only Y values

B has two polynomials of the same degree whose coefficients are defined over Zp and gives both polynomials to A. A picks a list of random X values, evaluates one of the polynomial and gives the result ...
1
vote
1answer
35 views

How to prove that the evaluation map is a ring homomorphism?

This is a really easy question, but I'm stuck in the logic of it... Let $F$ be an integral domain and $F[x]$ its polynomial ring. Let $a\in F$ fixed, define $\phi: F[x]\to F$ as ...
0
votes
1answer
32 views

Rational function in both $k(X)[Y]$ and $k(Y)[X]$

If I have a rational function in $X$ and $Y$ and it can be written as both a polynomial in $Y$ with coefficients being rational functions in $X$ (that is, an element of $k(X)[Y]$) and as a polynomial ...
2
votes
2answers
78 views

$2x+1 \in \mathbb{Z}_4[x]$ has no roots in any ring extension

Show that $f(x)=2x+1 \in \mathbb{Z}_4[x]$ has no roots in any ring $A$ that contains $\mathbb{Z}_4$. I don't know how to start here... If $\alpha$ is a root of $f$, then $2\alpha+1=0 \Rightarrow ...
3
votes
2answers
50 views

For which $p$ and $q$ polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $F_p[x]$?

It easy to prove that polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $\mathbb{Q}[x]$ if $(q,6)=1$, since they don't have a common zero in $\mathbb{C}$, this can be seen geometrically. My question ...
1
vote
3answers
82 views

Characterizing kernel of ring morphism

Let $K$ be a field and define a ring morphism $$\psi: K[x_1,x_2, \dots , x_n, y_1, y_2, \dots , y_n] \rightarrow K(x_1,x_2, \dots , x_n)$$ by $\psi(x_i) =x_i$ and $\psi(y_i) =\frac{1}{x_i}$. I ...
0
votes
0answers
17 views

Polynomial modulus in Quotient Ring

I have a ring $R=\Bbb Z[x]/(x^m+1)$ with $m$ some power of two and a polynomial $g \in R$, which has relatively small coefficients and some other properties that I believe to be irrelevant for this ...
2
votes
0answers
37 views

the numer of monic irreducible polynomials of degree $3$ in $\mathbb{F}_q$

I want to know how hany monic irreducible polynomials of degree $3$ there are in a field $\mathbb{F}_q$. The whole number of monic polynomials of degree three is $q^3$. Now I want to find out how ...
4
votes
2answers
55 views

Show that if $\mathrm{Tr}(y)=0$ then there exists a $x$ such that $x^p-x=y$.

We have the Trace map defined by: $$ \mathrm{Tr}\colon \mathbb{F}_q\rightarrow\mathbb{F}_q\colon x\mapsto x+x^p+x^{p^2}+\cdots+x^{p^{n-1}}, $$ where $q=p^n$. Now I have to prove that if ...
0
votes
0answers
24 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
1
vote
1answer
59 views

Characterizing maximal ideals in $\mathbb{Z}[x]$

I need to prove this: Let $I\subset\mathbb{Z}$ be the ideal generated by $\{p,f(x)\}$, with $p$ prime in $\mathbb{Z}$. Then $I$ is maximal iff $f(x)$ is irreducible modulo $p$. So I was trying to ...
1
vote
3answers
56 views

The number of ideals in the quotient ring $\mathbb R[x]/\langle x^2-3x+2 \rangle$ [duplicate]

Finding the number of ideals in the quotient ring $\mathbb R[x]/\langle x^2-3x+2 \rangle$. Attempt: $R[x]/\langle x^2-3x+2 \rangle = \{f(x)+\langle x^2-3x+2 \rangle~~|~~f(x) \in R[x]\}$. Since ...
5
votes
3answers
191 views

A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
0
votes
1answer
57 views

A Question about the Proof of Eisenstein's Irreducibity Criterion

Statement: Let $f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_0 \in \mathbb Z[x]$. If there is a prime $p$ such that $p \nmid a_n, p \mid a_{n-1}, \dots,p \mid a_0$ and $p^2 \nmid a_0 $, then $f(x)$ ...
0
votes
2answers
79 views

Is $2x^2+4$ reducible over $\mathbb C$?

I am not sure if I making some very fundamental mistake. But Gallian says that $2x^2+4$ is reducible over $\mathbb C$. If $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ is said to be ...
2
votes
0answers
56 views

Problems from Polynomial Rings. My attempt shown.

$1.$ Let $f(x) =a_mx^m+a_{m-1}x^{m-1}+ \cdots +a_0$ and $g(x) = b_nx^n+b_{n-1}x^{n-1}+ \cdots +b_0$ belong to $\mathbb Q[x]$ and suppose that $f \circ g \in \mathbb Z[x].$ Prove that $a_ib_j$ is an ...
3
votes
1answer
71 views

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field?

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field? My Thoughts: Suppose instead of $F$, we take the set of polynomials $R[x]$ over a commutative ring ...
5
votes
1answer
68 views

Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
2
votes
0answers
40 views

How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
3
votes
2answers
114 views

On factorization of polynomials

I would be very grateful if you give me a hint on problem 9, section 3.6 of Hungerford Algebra, regarding factorization in polynomial rings, saying that: Suppose $f(x)= ...
1
vote
2answers
18 views

Dimension of quotient construction

If I have an irreducible polynomial, $f$ with $deg(f) = n$ and I look at the quotient: $$R = \frac{\mathbb{Q}[x]}{(f)}$$ How can we show that the dimension of $R$ as a $\mathbb{Q}$ vector space is ...
2
votes
1answer
50 views

Facts about quotient rings - example

I have three quotient rings: $R_1 = \frac{\mathbb{Q}[x]}{(x^2 -1)}$ $R_2 = \frac{\mathbb{Q}[x]}{(x^2 +1)}$ $R_3 = \frac{\mathbb{Q}[x]}{((x -1)^2)}$ I am trying to decide whether these are integral ...
2
votes
2answers
64 views

Basic irreducible polynomial

I'm studying cyclic codes over a ring $R$. It is well known that a cyclic code over $R$ of length $n$ is an ideal of $R\left[ x \right]/\left( {{x^n} - 1} \right)$. Hence the factorization of ...
1
vote
1answer
18 views

Determining if any of these three are an ideal of $\mathbb{R}[x]$

$\mathbb{R}[x]$ denotes the ring of polynomials in $x$ with real coefficients. Let $I \subset \mathbb{R}[x]$ be the subset of those polynomials with constant coefficient $0$, and let $J \subset ...
1
vote
0answers
36 views

Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...
1
vote
2answers
55 views

roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
1
vote
1answer
27 views

On the existence of polynomial roots

Assume $F$ is a field, and $f\in F[x]$ is polynomial. To see that $f$ has a root in some extension of $F$, without loss of generality we can assume $f$ is irreducible. Indeed any polynomial $f$ is ...
2
votes
1answer
94 views

Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian?

Question: Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian? I guess it isn’t Noetherian as I suspect that $$ (x y + y^{2}), \quad (x y + y^{2},x^{2} y + ...
-4
votes
1answer
40 views

Question about nonisomorphic polynomial rings.

Let $n,k > 1$ be positive integers. Define the reduced polynomial rings : $g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is ...
1
vote
2answers
52 views

Question about kth root of a reduced ring element.

Let $n > 1$ be a positive integer. Let $k > 1$ be a positive integer. Define the reduced polynomial rings $f_n = \Bbb R[X_n]/(1+(X_n)^{n})$ How do we know if $(X_n)^{1/k}$ is an element of ...
0
votes
3answers
68 views

Primitive Root in Quotient Ring

Find a primitive root of $R[x]/\langle x^4+x+2 \rangle$ where $R$ is the integers mod $3$. Is there a good general stratagy to this sort of thing?
3
votes
2answers
109 views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = ...
0
votes
0answers
21 views

the vector space of canonical forms module I

let $I$ be an ideal of $K[X]$ and $S$ be the outside of the ideal generated by leading terms of $I$. How can I show that $K[X]$ is equal to direct sum of $I$ and $S$?
-4
votes
1answer
54 views

Can we abstractly construct this ring and what is it isomorphic to?

Define $R = \Bbb{Z}[X_1, X_2, \dots]$. Then place on $R$ the relations $$ X_1 + 1 = X_2, \\ X_2 + 2 = X_3, \\ X_3 + 2 = X_4, \\ X_5 + 2 = X_6, \dots \\ X_{2k-1} + 2 = X_{2k}, \ \forall k \geq 3 ...
0
votes
0answers
58 views

Different generators of (x,y) in k[x,y] give rise to automorphism.

I am stuck with the following algebra problem: Let $f,g\in k[x,y]$ be polynomials which generate $(x,y)$ (as an ideal). Consider the homomorphism $\phi:k[x,y]\to k[x,y]$ which is identity on $k$, and ...
2
votes
2answers
58 views

Nonzero nilpotent elements in $\Bbb C\otimes\Bbb Q[x]/(f)$?

I have to find if this affirmation is true: Let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$, then in $\mathbb{C}\otimes_{\mathbb{Q}} \mathbb{Q}[x]/(f)$ there are no nonzero nilpotent elements. ...
0
votes
0answers
33 views

Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greatest common divisor is 1) then $f(x)$ does not have multiple roots in $K$

Please I would like you to tell me if my proof is correct Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greates common divisor is 1) then $f(x)$ does not have multiple roots in $K$ ...
1
vote
0answers
56 views

Evaluation maps over polynomials

just looking for feedback and/or hints about this proof I've been working on. No answers please, but I'd like to know if I'm on the right track here. So I'm given a field $F$, and a non-zero $n ...
-1
votes
3answers
109 views

Quotient rings of polynomial rings

I have come across a quite difficult question while I am studying for a test: Let $F=\Bbb Z[x]/(7,x^2-3)$. Let $u$ denote the image of $x$ under the canonical epimorphism from $\Bbb Z[x]$ to ...
1
vote
1answer
15 views

Integral closure of a subring that is a polynomial ring over an algebraically closed field.

Let $K$ be an algebraically closed field that is a subring of an integral domain $D$. Assume $D$ contains an element $d$ that is transcendental over $K$. Also assume that $D$ is integral over ...
1
vote
1answer
45 views

Do polynomials make sense over non-commutative rings?

One could think of polynomials rings as sort of a derived ring (a ring of functions $f: \mathbb{N}^m \mapsto R$ such that $f^{-1}(R \setminus \{ 0 \} )$ is finite), but from what I can tell, we are ...
0
votes
1answer
30 views

Finding some homogeneous generators of an ideal.

Suppose that $\mathfrak a$ is an homogeneous ideal of $K[T_1,\ldots, T_n]$ where $K$ is a field of characteristic $0$ and $T_1,\ldots,T_n$ are indeterminates. Moreover suppose that $\mathfrak a$ has a ...
0
votes
2answers
38 views

Greatest common divisor of polynomials over $\mathbb{Q}$

I have two polynomials: $f: x^3 + 2x^2 - 2x -1$ and $g: x^3 - 4x^2 + x + 2$. I have to do two things: find $gcd(f,g)$ and find polynomials $a,b$ such as: $gcd(f,g) = a \cdot f + b \cdot v$. I have ...