# 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 $f(s)=0$.

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### Every element of a ring $A$ is integral over the ring $A$. How?

I am learning Integral Dependence for the first time. Every book says that the answer to my question is trivial but I don't see it. Please help!
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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)? ...
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### 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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### 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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### 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, ...
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### 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 $A$-...
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### 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.
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### 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 ...
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### 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 $B$). ...
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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 ...
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### $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ 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) \...
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### 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?
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### 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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### 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...
### 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 ...