4
votes
3answers
110 views

Polynomials over $\mathbb{F}_2$ with certain values in $\mathbb{F}_4$

Let $\mathbb{F}_4=\{0,1,u,u^2\}$ be the field with $4$ elements. Is there a polynomial $p \in \mathbb{F}_2[x,y]$ with the following property? (1) For $r,s \in \mathbb{F}_4$, we have $p(r,s)=u ...
1
vote
1answer
51 views

Factorization in noetherian domains

I changed the title (and the body) of this question page, since user26857 provided a nice answer for my original question in a more general setting. Here's what the accepted answer below provides: ...
3
votes
1answer
31 views

Basis-free and noncommutative versions of the two-polynomials-over-ring problem (McCoy theorem etc.)

There is a rather canonical bunch of exercises in commutative algebra which tend to come up time and again on math.stackexchange: recently in #948010 and #83121, formerly in #227787 and #413788, and ...
1
vote
1answer
69 views

Intersection of two polynomial ideals

In the $4$-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$ (I'm working on a algebraically ...
-1
votes
2answers
116 views

Show that every maximal ideal in $ \mathbb{Z}[x, y] $ contains a prime number [closed]

Let $\mathfrak{M} \subseteq \mathbb{Z}[x, y]$ be a maximal ideal. Show that $ \exists\ p \in \mathbb{Z}$, $p$ prime such $p \in \mathfrak{M}.$ Thanks for the answers. I'd be interested in a proof ...
0
votes
0answers
29 views

Kernel of homomorphism $A[X] \to B$ between integral domains [duplicate]

Let $A \leq B$ be integral domains, where $A$ is integrally closed and $B/A$ is an integral ring extension. Let further $\varphi : A[X] \to B$ be some homomorphism of $A$-algebras. Is the kernel ...
7
votes
1answer
69 views

The set of all $p \in \mathbb{C}[x]$ that can be expressed using only one occurrence of $x$.

Let $X$ denote the least subset of $\mathbb{C}[x]$ subject to the following constraints. $x \in X$. $p \in X \rightarrow ap \in X,$ for all $a \in \mathbb{C}$. $p \in X \rightarrow p+a \in X,$ for ...
0
votes
0answers
23 views

Characterization of ideals generated by homogeneous polynomials in terms of $f^{(d)}$ in Gathmann's notes.

On pg. 37 of Gathmann's Algebraic Geometry notes, the following is mentioned: For every $f\in k[x_0,x_1,\dots,x_n]$ be an ideal. The following are equivalent: I can be generated by ...
5
votes
4answers
192 views

Euclid's proof of the infinitude of primes to prove this question

I'm trying to prove that if $k$ is a field, then there are an infinite number of irreducible monic polynomials in $k[X]$. My attempt of solution is use almost the same strategy of the Euclid's proof ...
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, ...
0
votes
0answers
54 views

Proof that $\mathbb{Z}[\sqrt{-5}]$ is integrally closed

There are demonstrations on the Internet saying that the polynomial $$\left(x-\frac{a}{c}-\frac{b}{d}\sqrt{-5}\right)\left(x-\frac{a}{c}+\frac{b}{d}\sqrt{-5}\right)$$ is monic if and only if ...
2
votes
1answer
79 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
2
votes
1answer
95 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 + ...
0
votes
1answer
55 views

Powers generate monomials

What is a reference in the literature for the following fact? Let $A$ be a commutative $\mathbb{Q}$-algebra. Then every monomial in $A$ of degree $n$ may be written as a linear combination of $n$th ...
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 ...
1
vote
2answers
52 views

There are elements in $\mathbb Z [t]$ such that this sum is 1

In order to solve a problem I'm facing I want to prove that there are $f_1,f_2,f_3$ elements in $\mathbb Z[t]$ such that $f_1(t)\cdot(4t-4)+f_2(t)\cdot(5t)+f_3(t)\cdot(t^2-17)=1$. In another words, I ...
3
votes
4answers
145 views

Finding generators for an ideal of $\Bbb{Z}[x]$

We know that $\Bbb{Z}$ is Noetherian. Hence, we can conclude that $\Bbb{Z}[x]$ is Noetherian, too. Consider the ideal generated by $\langle 2x^2+2,3x^3+3,5x^5+5,…,px^p+p,…\rangle$ for all prime ...
0
votes
1answer
31 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
0answers
89 views

Vanishing polynomials

Let $K$ be a field and $V$ be the set of points $(t^3,t^4,t^5)$ where $t$ is in $K$. Set $I=(Y^2-XZ,Z^2-X^2Y,X^3-YZ)$. Show that $I$ is a subset of $A$, where $A$ is the set of polynomials which ...
1
vote
0answers
91 views

Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
1
vote
0answers
104 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
8
votes
2answers
548 views

How does Hilbert's Nullstellensatz generalize the “fundamental theorem of algebra”?

What is Hilbert's Nullstellensatz in the sense of the generalization of "fundamental theorem of algebra"? I've seen that in some texts it was referred to as the generalization of the fundamental ...
5
votes
1answer
56 views

Topological closure of ideal in $A[[T]]$ - Proposition 1.3.7 in Liu

In Proposition 1.3.7 of Liu's book, one proves that if a ring $A$ is noetherian then so is $A[[T]]$. We take an ideal $I$ of $A[[T]]$ and prove that there exist $F_1,\ldots,F_m\in I$ such that for all ...
0
votes
0answers
62 views

Homogeneous polynomial in a homogeneous ideal

Let $f$ be a non-zero homogeneous polynomial in a homogeneous ideal generated by homogeneous elements $g_1,\ldots, g_s$. Suppose $f= h_1g_1 +\cdots+h_sg_s$. Is it necessary that $\deg(f)=\deg(h_ig_i)$ ...
2
votes
1answer
69 views

Projecting an affine hypersurface away from a point in its projective closure is never a finite map?

Let $X\subset \mathbb{A}_k^r$ be an irreducible hypersurface defined by a polynomial $g$, where $k$ is an algebraically closed field. Embed $\mathbb{A}^r\hookrightarrow\mathbb{P}^r$ in the usual way. ...
2
votes
2answers
120 views

Help in this proof in Lang's Algebra book

I'm trying to understand this part of the proof: I didn't understand why not all coefficients of $f_2,\ldots,f_n$ can lie in the maximal ideal, maybe I'm forgetting something, it should be a very ...
0
votes
0answers
79 views

Multivariable irreducible polynomials over finite fields

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
1
vote
1answer
53 views

Why are $(X_1), (X_1,X_2), \ldots$ prime ideals?

I was looking at the proof of the dimension of the polynomial ring $R[X_1,\ldots,X_n]$ and I had a question: Why are $(X_1), (X_1,X_2), (X_1,X_2,X_3),\ldots, (X_1,\ldots,X_n)$ prime ideals in this ...
0
votes
1answer
37 views

Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
2
votes
1answer
84 views

Showing that if $f,g \in k[x,y]$ are irreducible and not associates then $(f,g) \cap k[x] \ne 0$

There is a part of example 10.25.3 at http://stacks.math.columbia.edu/tag/00EX that I'm having trouble understanding. Here, $k$ is a field and $f,g \in k[x,y]$ are irreducible and are not associates. ...
2
votes
1answer
90 views

Calculating Grobner Bases

In this question, $ℚ[x,y,z]$ is endowed with the lexicographic order with $x > y > z$. Set $u:= x^2 + 2yz^2$ and $v:= y^2 - 3xz$. Denote by $J$ the ideal of $ℚ[x,y,z]$ generated by $u$ and $v$. ...
2
votes
1answer
110 views

Monic irreducible polynomial over an integral domain

These days, I have some basic problem in abstract algebra. I know that in any integral domain, any prime element must be an irreducible element. Moreover, if $A$ is a UFD, then an element $a \in A$ is ...
0
votes
0answers
108 views

A question in Chapter III.4 of Dino Lorenzini's “An Invitation to Arithmetic Geometry”

Question 1 I am studying in the book "An Invitation to Arithmetic Geometry" by Prof. Dino Lorenzini. In Chapter III Section 4, we consider the following condition: Let $A$ be a Dedekind domain ...
4
votes
2answers
134 views

Exercise from Atiyah-Macdonald, Chapter 1, 2.iv)

Let $A$ be a ring and let $A[x]$ be the ring of polynomials in an indeterminate $x,$ with coefficients in $A.$ Let $f=a_0 + a_1x+\cdots+a_nx^n \in A[x].$ $f$ is said to be primitive if ...
1
vote
2answers
337 views

Homogeneous polynomial in $k[X,Y,Z]$ can factor into linear polynomials?

My question is quite simple. Let $k$ be a closed algebraic field and $f\in k[X,Y]$. We know that $f$ can factor into linear polynomials. I would like to know if there is some generalization of ...
3
votes
2answers
123 views

What is the proof of the single factor theorem over an arbitrary commutative ring?

Theorem (Single factor theorem) Let $R$ be a commutative ring, and let $P\in R[X]$, where $R[X]$ is the polynomial ring over the indeterminate $X$. Suppose $P(\alpha)=0$. Then $(X-\alpha)$ divides ...
0
votes
1answer
82 views

Kernel and direct sum

Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ...
0
votes
1answer
166 views

Subrings of polynomial rings over the complex plane

I have the following questions: (i) must every subring of the polynomial ring in two variables over the complex plane, containing the complex plane itself, be Noetherian? (ii) Are there Noetherian ...
0
votes
0answers
89 views

What is $Spec(\mathbb{Z}[x])$? [duplicate]

What is $Spec(\mathbb{Z}[x])$? For a commutative ring $A$ e with $1$, its spectrum $Spec(A)$ is defined to be the set of all of its prime ideals. So the question is to find all the prime ideals of the ...
0
votes
0answers
81 views

$f$ is irreducible iff $V(f)$ is irreducible

I would like to know if the following statement is true: $f$ is irreducible iff $V(f)$ is irreducible. My tools I'm trying to use to prove this are Study's Lemma and basic algebra. If $f$ is ...
1
vote
1answer
47 views

The height of a prime ideal in the $\kappa[[X]][Y]$

Let $\kappa$ be a field and $S=\kappa[[X]]$ be the ring of power series which depends on the indeterminate $X$. Now consider the ring $S[Y]$, the ring of polynomials with coefficients in $S$ and ...
5
votes
1answer
90 views

Algebraic independence in $ k[x,y]$

Let $k$ be a field, then $x$ and $y$ are algebraically independent in polynomial ring $k[x,y]$, so I would guess that 2 is the maximal number of algebraically independent elements in $k[x,y]$ But I ...
4
votes
2answers
232 views

Nilpotent/invertible polynomial over commutative ring. [duplicate]

Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial over a commutative ring $R$. Prove that (a) $p$ is unit in $R[x]$ iff $a_0$ is unit and $a_1,a_2,\ldots,a_n$ are nilpotent in ...
2
votes
1answer
49 views

isomorphic quotient rings of polynomial ring and Hilbert functions

Let $k$ be a field, $R=k[x_1,\cdots,x_n]$ and $I,J$ homogeneous ideals of $R$. Denote by $H_I(s), H_J(s)$ the Hilbert functions of $I,J$ respectively. If $R/I, R/J$ are isomorphic as graded rings, ...
5
votes
1answer
152 views

What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
2
votes
2answers
124 views

What is the ideal of leading terms?

Fix a monomial ordering on the polynomial ring $\Bbb{k}[x_1, \dots, x_n] = R$ over a field. What exactly is $LT(I)$ for an ideal $I$ of $R$? How is it defined and does it form an ideal?
4
votes
1answer
100 views

Random algebraic numbers are linearly disjoint almost surely?

It is well-known that if one considers a “random” monic polynomial of fixed degree, say $X^n+\sum_{k=0}^{n-1}a_kX^k$ where $(a_0,a_1,\ldots, a_n)$ is drawn from the discrete uniform distribution on ...
1
vote
0answers
20 views

General position for one-parameter family of algebraic numbers

Let $P(x,y)$ be an irreducible twovariate polynomial with rational coefficients such that $P(n,.)$ has degree $>1$ for any $n\in{\mathbb N}$. For any $n\in{\mathbb N}$, one may choose a root ...
4
votes
2answers
168 views

Can $\operatorname{Spec}(R[X])$ ever be finite?

I have a quick question. Suppose $R$ is a nonzero commutative ring. Is it possible that $|\operatorname{Spec}(R[X])|<\infty$?
2
votes
2answers
93 views

Is the ring of polynomial invariants of a finite perfect group an UFD?

Let $G$ be a finite group. $G$ acts on $\mathbb K[x_1,...,x_n]$ by automorphisms fixing $K$. $\mathbb K[x_1,...,x_n]^G=\{ T\in \mathbb K[x_1,...,x_n],\forall \sigma \in G, T^{\sigma}=T\}$ is the ring ...