1
vote
1answer
41 views

GCD for multivariable polynomial ring

I'm reading Lectures on Modules and Rings by T. Y. Lam. It's on page 32 of the book, example 2.19A. It reads: (2.19A) Example. Let $k$ be a field. Then in the commutative polynomial ring $R = ...
-2
votes
4answers
85 views

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$. How to prove? I really have no idea... Thank you a lot.
1
vote
1answer
38 views

irreducible elements of polynomial rings

Let $p$ be a prime integer. For $x\in\mathbb{Z}$, let $x'$ be the remainder of $x$ when divided by $p$. Let $\sum_{i=0}^{n}a_iX^i\in \mathbb{Z}[X]$ with $p$ does not divide $a_n$ in $\mathbb{Z}$. Then ...
1
vote
1answer
24 views

If $u$ and $v$ have the same order, $u=f(\tau)(v)$ iff $v=g(\tau)(u)$?

Suppose $\tau\in L(V)$, for $V$ a finite dimensional vector space, and $p(x)$ is an irreducible factor of the minimal polynomial $m_\tau(x)$. Also, suppose $u,v\in V$, and both $u$ and $v$ has order ...
8
votes
2answers
105 views

$R$ with an upper bound for degrees of irreducibles in $R[x]$

One very convenient property of $\mathbb{R}$ as a ring is that there is an upper bound for the degree of irreducible polynomials in $\mathbb{R}[x]$, as If $f\in\mathbb{R}[x]$ has degree larger ...
1
vote
1answer
118 views

Property of an operator in a finite-dimensional vector space $V$ over $R$

Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$. For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$. (a) Prove that there exists $N$ such that ...
0
votes
3answers
294 views

$R = \mathbb{Z}[ i ] / (5)$ is not an integral domain? Why?

Let $R = \mathbb{Z}[ i ] / (5)$ . How should I prove that $5 = (2+i) (2-i)$ is a prime factorization in $\mathbb{Z}[i]$? Can we deduce from this that R is not an integral domain? How? I know that ...
1
vote
1answer
125 views

Is $R = Q[x] / (x^4 - 3x^2+ 6x)$ isomorphic to a direct sum of two fields?

Let $R = Q[x] / (x^4 - 3x^2+ 6x)$. How can we prove that $x^2 + 1$ is invertible in $R$? How can we prove that $R$ is isomorphic to a direct sum of two fields?
10
votes
4answers
256 views

Show $\mathbb{Q}[x,y]/\langle x,y \rangle$ is Not Projective as a $\mathbb{Q}[x,y]$-Module.

Disclaimer: Though I have been re-reading my notes, and have scanned the relevant texts, my commutative algebra is quite rusty, so I may be overlooking something basic. I want to show $\mathbb{Q} ...
5
votes
2answers
347 views

On rank of a matrix whose entries are polynomials

(I took courses on linear algebra, but I don't know anything about $R$-modules or such things.) How do you define the rank of a matrix whose entries are polynomials in $K[X]$? If you assign some ...
4
votes
1answer
301 views

The ideal $(x,y)$ is not a free $K[x,y]$-module

Given a field $K$ we have the polynomial ring $K[x,y]$ in $2$ variables, which is also a left module (over itself). How can we prove that the ideal $(x,y)$ is not a free module?
0
votes
1answer
112 views

What is the algorithm to compute the kernel of a module homomorphism?

http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/02/paper_html/node27.html What is the algorithm to calculate the kernel of a module as defined in the link above? According to my understanding, ...
2
votes
1answer
136 views

Basis of a submodule of $\mathbb{Q}[X]^3$ over $\mathbb{Q}[X]$

I want to find the basis of a submodule of $\mathbb{Q}[X]^3$ generated by $(2X-1, X, X^2+3)$, $(X,X,X^2)$ and $(X+1,2X,2X^2-3)$. I determine that $$ \begin{pmatrix} 2X-1 & X & X^2+3 \\ X ...