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Questions tagged [integral-dependence]

Integral dependence (also known as algebraic dependence) is a condition on ring extensions $R\subseteq S$: we say $s\in S$ is integral over $R$ if there is some $f(x) \in R[x]$ such that $f(s)=0$.

Integral dependence (also known as algebraic dependence) is a condition on ring extensions $R\subseteq S$: we say $s\in S$ is integral over $R$ if there is some $f(x) \in R[x]$ such that $f(s)=0$. The case where $R,S$ are fields forms the starting point of much of Galois theory.

An equivalent condition: $s\in S$ is integral iff there is a subring $T\subseteq S$ containing $s$ that is finitely-generated as an $R$-module. The idea is to use the (finite) powers of $s$ in $f(s)$ as generators.

Consider using tags such as , , , or similar in conjunction with this tag.

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Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$

This seems to be one of those tricky examples. I only know one proof which is quite complicated and follows by localizing $\mathbb{Z}[\sqrt[3]{2}]$ at different primes and then showing it's a DVR. Does anyone know any simple quick proof?
pki
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UFDs are integrally closed; so too are GCD & Dedekind domains.

Let $A$ be a UFD, $K$ its field of fractions, and $f$ an element of $A[T]$ a monic polynomial. I'm trying to prove that if $f$ has a root $\alpha \in K$, then in fact $\alpha \in A$. I'm trying to exploit the fact of something about irreducibility,…
James R.
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Proposition 5.15 from Atiyah and Macdonald: Integral Closure and Minimal Polynomial

I am having some trouble understanding Proposition 5.15 in Introduction to Commutative Algebra by Atiyah and Macdonald. Let $A\subset B$ be integral domains, $A$ integrally closed, and $x\in B$ be integral over an ideal $\mathfrak{a}$ of $A$. Then…
kfriend
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Algebraic vs. Integral Closure of a Ring

Let $R\subseteq S$ be a ring extension. It is true that the set of elements of $S$ that are are integral over $R$ (i.e. satisfy a monic polynomial equation over $R$) is a subring of $S$. Can anyone provide an example showing that the set of…
Drew Armstrong
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Working out the normalization of $\mathbb C[X,Y]/(X^2-Y^3)$

I'm trying to identify the normalization of the ring $A := \mathbb C[X,Y]/\langle X^2-Y^3 \rangle$ with something more concrete. First, $X^2-Y^3$ is irreducible in $\mathbb C[X,Y]$, making $\langle X^2-Y^3\rangle$ prime, so $A$ is a domain and it…
Pece
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Normalization of a quotient ring of polynomial rings (Reid, Exercise 4.6)

I solved all parts of Exercise 4.6 of the book Undergraduate Commutative Algebra of Miles Reid except the last one. Let $A=k[X]$ and $f\in A$ has a square factor but it is not a square polynomial itself, then what will be the normalization of…
H.W.
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Normalisation of $k[x,y]/(y^2-x^2(x-1))$

I am trying to figure out the normalisation of $k[x,y]/(y^2-x^2(x-1))$, for an algebraically closed field $k$. I can show that it is not normal and I have the information that the normalisation is $k[t]$, where $t=\frac{y}{x}$. I can't figure out…
baltazar
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$k[X]$ is integral over $k[X^{2}]$

I am trying to show that $k[X]$ is integral over $k[X^2]$, where $k$ is a field. Taking an element $b=b_nx^n+b_{n-1}x^{n-1}+...b_1x+b_0 \in K[X]$ we want to find $a_i \in K[X^2]$ such that $a^nb^n+a_{n-1}b^{n-1}+...a_1b+a_0=0$. I am stuck because…
user127214
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Hints about Exercise 4.2 in Miles Reid, Undergraduate Commutative Algebra

$ A \subset B $ is a ring extension. Let $ y, z \in B $ elements which satisfy quadratic integral dependance $ y^2+ay+b=0 $ and $ z^2+cz+d=0 $ over $ A $. Find explicit integral dependance relations for $ y+z $ and $ yz $. It is given this hint…
Riccardo
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Is the integral closure of an integrally closed Noetherian domain in a finite extension field Noetherian?

Just as the title says. Let $R$ be a Noetherian integral domain, let $K$ be its field of fractions, let $L$ be a finite extension of $K$, and let $S$ be the integral closure of $R$ in $L$. Must $S$ be Noetherian, or do I need some additional…
calearner
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Integral ring extension

Let $R=k[t]/(t^2)$ and $S=k[t,x]/(t^2,tx^3+tx^2-x^2-x)$, $k$ is a field. I must prove that $S$ is integral over $R$ and that $S=R\oplus R$. Any help about that..thanks in advance...
georgeson1960
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Is the preimage of the non-normal locus a divisor?

Let $X$ be a complex, affine variety. Let $\nu:\tilde X\to X$ be the normalization of $X$ and denote by $D\subseteq X$ the closed set of points where $\nu$ fails to be an isomorphism, i.e. $D$ is the non-normal locus of $X$. Question 1. Under which…
Jesko Hüttenhain
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Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$

Let $A$ and $B$ be two commutative rings with a unit element, with $B$ subring of $A$. Suppose $x$ is an invertible element in $A$. Then prove that the intersection of the two rings $B[x] \cap B[x^{-1}]$ is integral over $B$, i.e., prove that for…
Li Xinghe
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Quotient Rings and Integral Extensions

Suppose $S$ is an integral extension of $R$ and $I$ an ideal in $S$. Why is $S/I$ an integral ring extension of $R/(R \cap I)$? To this question, Dummit and Foote says: Reducing the monic polynomial over $R$ satisfied by $s \in S$ modulo $I$…
user1205
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Finding the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$

I've just learned what the integral closure is. I would like to find what is the intergral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$. Let $\mathcal{R}$ the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$. To determine $\mathcal{R}$ I started…
Zanzi
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