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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Show that $\alpha_i$ are integral over $k[x]$ and generate the integral closure over $k[x]$.
Let $k$ be a field of characteristic $2$. Let $f(x) \in k[x]$ be a polynomial of positive degree without double roots in $k$.
Also assume that $k$ is algebraically closed and perfect.
Then $f'(x)$ is ...
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Intuitive Meaning of Integral Dependence
I've recently came across the notion of a Ring $B$ being integral over a subring $A$. Atiyah-MacDonald (at least the few pages right after the definition) refrain from explaining the intuitive ...
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Show that $\mathbb Z[t^4,t^7]$ is not integrally closed in $\mathbb Z[t]$
To show $\mathbb Z[t^4,t^7]$ is not integrally closed in $\mathbb Z[t]$ is it fine to say that $t$ is a root of $x^4-t^4$ which is a monic polynomial with coefficients in $\mathbb Z[t^4,t^7]$, but $t\...
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Coefficient of integral dependence and prime ideals
Let $A \subseteq B$ be an integral extension of rings. Let $b \in B$, and let $b^n+a_1b^{n-1}+\dots+a_{n-1}b+a_n=0$ be an integral dependence relation for $b$, with $a_1,\dots,a_n\in A$.
Given a prime ...
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Integral Dependence
A module theoretical equivalence of integral dependence states as follows:
Let $S$ be a ring and $R \subseteq S$ be a subring, and $1\in R$. Then $x \in S$ is integral over $R$ if and only if $R[x]$ ...
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Let $A\subset B$ be an integral ring extension, and $p\in\operatorname{Spec}A$. Then there exists $q\in\operatorname{Spec}B$ such that $q\cap A=p$.
I need an example of this theorem. I could think of trivial example if $B=A$. But any suggestions for a non trivial example?
How is this theorem true in case $A = \mathbb C[x,y]/\left<y^2-x^3\right&...
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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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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}(...
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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 $\...
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