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Questions tagged [integral-dependence]

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1answer
44 views

Generalization of Atiyah-Macdonald Proposition 5.7

The Proposition is Let $A\subseteq B$ be integral domains, $B$ integral over $A$. Then $B$ is a field iff $A$ is a field. The proof is easy. I want to generalize this proposition. I want to prove ...
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0answers
13 views

Long Range Dependence, Order Statistics

I have a long range dependence process which is defined in such a way: $$σ^2_{μ,X}=∫_T∫_{R^2}Cov_X(t,u,v) μ(du) μ(dv) dt. $$ We can use derived formula for covariation using distribution functions ...
0
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0answers
69 views

How to solve analytically this equation?

I need to solve the next equation and to find $x_1$: $$\dfrac{d(x_0x_1)}{dz}=-8\dfrac{dx_0}{dz}\dfrac{1}{r-z(r-1)}$$ where $x_0$, $x_1$, $z$ are variables, and $r$, $\beta$ are constants. At this ...
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2answers
89 views

How to solve multiple dependent differential equation?

I have next differential equation: $$ (x_0x_3)'=-(x_1x_2)'+8\cdot(-x_2'-\dfrac{x_1'}{x_0}+\dfrac{x_1x_0'}{x_0^2})$$ where $x_0'=\dfrac{dx_0}{dz}$, and it means the same for every sign $'$. On the ...
1
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1answer
54 views

Integration boundary condition dependent on integral derivative?

I need to solve integral: $\int_0^{r(z)} \dfrac{\mathrm{d}z}{r^4(z)-2r^2(z)}$, where $r(z)=r_i-z(r_i-1)$, where $r_i$ is constant, $z$ is longitudinal coordinate. Boundary condition $r(z)$ is ...
0
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1answer
337 views

Going Up Theorem - Examples? Witnesses?

I am currently in Chapter 5 - Integral Dependence and Valuations of the text Introduction to Commutative Algebra by Atiyah - Macdonald. I am in particular studying the `Going-up theorem', and I have a ...
4
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1answer
47 views

Domain without any non trivial integral extensions (any monic polynomial $f \in A[x] \setminus \{1\}$ has a root in $A$) is a field?

Let $A$ be an integral domain. Assume that any monic polynomial (different from $1$) with coefficients in $A$ has a root in $A$. Does it follow that $A$ is a field (necessarily algebraically closed)? ...
1
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1answer
122 views

Let $k$ be a field of characteristic $\neq 2$, and $n\geq m\geq 3$. Then $k[x_1,\ldots,x_n]/\langle x_1^2+\ldots+x_m^2\rangle$ is integrally closed. [duplicate]

I wish to show that $B:=k[x_1,\ldots,x_n]/\langle x_1^2+\ldots+x_m^2\rangle$ is a normal domain -- that is to say, it is integrally closed in its ring of fractions. Unfortunately, I don't know of any ...
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3answers
199 views

Integral closure of Gaussian Integers

I am considering $\mathbb{Z}[i]\subset\mathbb{Q}(i)$ Now I have a short note here that says that there are elements of $\mathbb{Q}(i)$ which are not integral over $\mathbb{Z}[i]$ 'because $\mathbb{Q}(...
5
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1answer
265 views

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 $\...
1
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1answer
192 views

What is the integral closure of the integers in the real numbers?

What is the integral closure of the ring $\mathbb{Z}$ inside the field $\mathbb{R}$ of real numbers and what are it's properties? Is this studied at all?
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0answers
557 views

How to compute the integral closure of $\Bbb{Z}$ in $\mathbb Q(\sqrt[n]{p})$?

We have the definition of integral closure that all the integral elements of A in B. Could we just compute the integral closure of certain A in B. I am considering such a problem that given a prime p, ...
2
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1answer
164 views

Equivalent conditions for integral element

Let $A\subset B$ be a ring extension and $x\in B$. Then if $A$ is noetherian the following two conditions are equivalent: $i)$ $x$ is integral over $A$. $ii)$ There exists a finitely generated $A$-...
3
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1answer
301 views

$x$ is integral over $R$ if and only if for every minimal prime $\mathfrak q$ of $S$, $x$ is integral over the residue domains

I've made some progress on the following problem: Let $R$ be a Noetherian ring, $R \subseteq S$ an extension of rings, and $x \in S$. Show that $x$ is integral over $R$ if for every minimal prime $...
1
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1answer
288 views

Ring of fractions of an integrally closed integral domain is also integrally closed

I hope that the following questions can be solved by Ring Theory concepts as I'm not studying further yet. Let $A$ be an integral domain and $K$ its field of fractions. An integral domain $A$ is said ...
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3answers
145 views

Gaussian Integer is Integral Over Integer

I am reading from Wikipedia's article on Integral Element here, where it says that, ... an element $b$ of a commutative ring $B$ is said to be integral over $A$, a subring of $B$, if there are $n \...
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1answer
54 views

Find a monic polynomial [closed]

For each $f \in K[x]$, find a monic polynomial satisfied by $f$ with coefficients in $K[x^2]$, where $K$ is a field.
5
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1answer
156 views

Show that $A[x] \cap A[x^{-1}]$ is integral over $A$. [duplicate]

Let $R$ be a commutative ring, $A$ a subring of $R$, and $x$ a unit in $R$. Show that every $y \in A[x] \cap A[x^{-1}]$ is integral over $A$. I'm supposed to use the fact that there exists an ...
2
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1answer
56 views

Why do we need injectivity in the definition of integral dependence?

Let $f: A \rightarrow B$ a ring morphism of commutative rings, then one has on $B$ a multiplication by elements of $A$ defined by $b*a \doteq b.f(a)$ (where . is the multiplication in the ring $B$). ...
4
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2answers
625 views

Integral closure of $\mathbb{Z}$ in $\mathbb{C}$ is not finitely generated as a $\mathbb{Z}$-module?

Let $$ \mathbb{Z}^{'}_{\mathbb{C}}=\{ z \in \mathbb{C} | \exists f \in \mathbb{Z}[X] \text{ monic such that } f(z)=0\} $$ be the integral closure of $ \mathbb{Z} $ in $ \mathbb{C} $. Prove that $ \...
2
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2answers
149 views

Extending an automorphism to the integral closure

I need some help to solve the second part of this problem. Also I will appreciate corrections about my solution to the first part. The problem is the following. Let $\sigma$ be an automorphism of ...
1
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2answers
64 views

Show that $B/Q$ is integral over $A/P$

If $A$ is a subring of $B$ and $B$ is integral over $A$, let $Q$ be a prime ideal of $B$ and $P=Q\cap A$. Show that $B/Q$ is integral over $A/P$. If $b\in B$ is integral over $A$ then for some ...
5
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0answers
106 views

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 ...
1
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1answer
45 views

Which type of correlation should I use?

I am beginner in statistics. I have excel table with few columns. I would like to find correlation between the variables. I have to make an essay to my boss and he wants concrete answers. I searchin ...
4
votes
1answer
153 views

Extension of Integral Domains

Let $S\subset R$ be an extension of integral domains. If the ideal $(S:R)=\{s\in S\mid sR\subseteq S\}$ is finitely generated, show that $R$ is integral over $S$. My first attempt was to show that $R$...
2
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2answers
203 views

Integral extension is a finitely generated $R$-module?

Let $R$ be a commutative ring. If $b_1,\ldots,b_n$ are elements of a ring $R'$ (commutative) which are integral over $R$ then $R[b_1,\ldots,b_n]$ is a f.g. $R$-module. My question is: If $\{b_i\}_{...
6
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1answer
855 views

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 $...
2
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1answer
51 views

$F/E$ a finite Galois extension, the integral closure of $E[X]$ in $F(X)$

If $F/E$ is a finite Galois extension, then one can show that $F(X)/E(X)$ is also a finite Galois extension of the same degree (a basis for $F/E$ is also a basis for $F(X)/E(X)$). Since $E[X]$ is a ...
2
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1answer
71 views

Integral extension and field

I came a cross a question that I don't know how to solve Problem: $A,B$ are commutative domains and $A\subseteq B$. Show if that $B$ is a field and every element of $B$ is the root of a non-trivial ...
1
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1answer
127 views

Are locally integral extensions integral? [closed]

Let $R$ be a commutative ring with unit and let $A \subseteq B$ be commutative $R$-algebras. Suppose $B_P$ is integral over $A_P$ for all primes $P$ of $R$. Is $B$ then necessarily integral over $A$ ?
4
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1answer
440 views

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 ...
3
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0answers
733 views

Difference between algebraic and integral extension

I have been reading Miles Reid Undergraduate Commutative Algebra and in chapter 4 he talks about a crucial difference between algebraic extension and integral extension (see the picture below). Now I ...
2
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1answer
51 views

Integral dependence and field extension

Let $R$ be a domain (commutative with unity). $k$ is field algebraically dependent on $k_0$. $A$ is some ideal of $R \otimes_{k_0} k$ and $A_0$ = $A \cap R$. How to prove that $(R \otimes_{k_0} k)/A$ ...
3
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1answer
211 views

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}...
8
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1answer
839 views

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 ...
2
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2answers
611 views

$\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is integrally closed

I'm was browsing this question, where it is proven the quotient field of $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is isomorphic to the rational function field $\mathbb{Q}(t)$ under the isomorphism $$ (x,y) \...
2
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1answer
69 views

Dimensions of integral ring extensions

If $X$ is a commutative ring with identity and $Y$ is an integral $X$-algebra, show that $\dim\,X=\dim\,Y$. I think also that $X$ needs to be a subring of $Y$. Why is this true?
2
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1answer
71 views

Simple integral extension question

If $R$ is a commutative ring, why is every $x$ in $R$ integral over $R$? I can't see what monic polynomial will have $x$ as a root.
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2answers
357 views

$k[X]$ 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+...
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1answer
1k views

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 ...
0
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1answer
110 views

Ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$

Let $p,q$ be distinct odd primes, and $\omega_p,\omega_q$ primitive $p$-th, $q$-th roots of unity. What is the ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$? The numbers in the ring $\mathbb{Q}(...
2
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2answers
525 views

Integral dependence over rings is transitive

Let $A\subset B\subset C$ be commutative rings. Suppose $B$ is integral over $A$, and $C$ is integral over $B$. Then I want to show that $C$ is integral over $A$. To be integral means that for every $...
7
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1answer
115 views

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...
1
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1answer
199 views

Is the ring $\mathbb{C}[t^2,t^3]$ integrally closed?

I am trying to understand if the ring $\mathbb{C}[t^2,t^3]$ in integrally closed (into its field of fractions), but I have no idea about how to proceed. All I have tried until now has failed. Any ...
1
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0answers
60 views

Show that $\operatorname{Spec}k[x_1,x_2,…,x_n]/(x_1^2+\cdots+x_m^2)$ is normal for $\operatorname{char}k\neq 2, n\ge m \ge 3$ [duplicate]

I want to show that if $F(T) \in B[T]$, where $B:=k[x_1,x_2,...,x_n]/(x_1^2+\cdots+x_m^2)$, is monic and has a root $\alpha \in\mathcal K(B)$ then $\alpha$ actually lives in $B$. This will imply that $...
3
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2answers
124 views

If $x$ is integral over $A_m$ for all maximal ideals $m$, then $x$ is integral over $A$

I am going over an old exam, and there is this question that I am stuck: Given $A$ a commutative ring with unity, show that if $x\in\operatorname{Frac}(A)$ is integral over $A_m$ for all maximal ...
2
votes
4answers
306 views

Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then $x$ is not integral over $S$

Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then I want to show that $x$ is not integral over $S$ I'm not seeing why this is the case....
1
vote
1answer
68 views

Necessity of an hypothesis in this exercise about integral dependence of $k[x][\frac{1}{f}]$ over $k[x]$

let $k[x]$ be the ring of polynomial over a field $k$. Let $f \in k[x]$ an irreducible polynomial. Then $k[x][\frac{1}{f}]$ is not integral over $ k[x]$. I solved this ex. In this way By absurd, $\...
8
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1answer
596 views

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 ...
6
votes
1answer
2k views

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^...