# Tagged Questions

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

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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### Reduced Gröbner basis and extension of scalars

Consider a field extension $L\subseteq K$, and let $\mathfrak a\neq 0$ be an ideal of the polynomial ring $L[T_1,\ldots,T_n]$. Suppose that a monomial order is fixed, so there exists a unique reduced ...
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### 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 ...
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### 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" ...
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### 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 + ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
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### 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 ...
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### 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 ...
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### 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 $\deg(f)=\deg(h_ig_i)$ ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $f$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### Can $\operatorname{Spec}(R[X])$ ever be finite?

I have a quick question. Suppose $R$ is a nonzero commutative ring. Is it possible that $|\operatorname{Spec}(R[X])|<\infty$?
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### 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 ...
### If $X$ is a cone, show that $I(X)$ is homogeneous.
The exercise is 1.3(3) from HP Kraft, "Appendix A: Basics from Algebraic Geometry." If a closed subset $X\subseteq \mathbb C^n$ is a cone, show that $I(X)$ is generated by homogeneous functions. ...
### Any finite set in $k^n$ is an algebraic set.
I'm trying to show that given a field $k$, and a finite set of points $\{a^i: i = 1\dots n\} \subset k^n$ is an algebraic set or equivalently is the set of common zeros of some set of polynomials \$S ...