# Questions tagged [integral-extensions]

This tag is for questions regarding to the Integral Extensions or, Integral Ring Extensions.

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### integral extension and existence of finitely generated faithful module

I am reading the book Advanced modern algebra by Rotman. In the integral extension section, he claims that if $R^*/R$ is a ring extension and $u\in R^*$ be a nonzero element then If there is a ...
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### Galois group of $x^5-x-1$ over $\Bbb Q$ using integral extension ring theory

Let $f= x^5 − x − 1$ and $L$ be the splitting field of $f(x)$ over $\Bbb Q$. Suppose $B$ is an integral closure of $\Bbb Z$ in $L$ and $P$ is a maximal ideal of $B$ such that $P \cap \Bbb Z = 2\Bbb Z$...
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### If a finite group acts on an integral domain which is integrally closed, then the fixed point subring is also integrally closed

Let $G$ be a finite group acting on an integral domain $R$, and let $S$ denote the fixed point subring: $$S=\{x\in R: gx=x \text{ for all }g\in G\}$$ I am asked to show that $R$ is integral over $S$, ...
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### On the “largest ring extension” of a Noetherian domain inside fraction field that is module finite

Let $R$ be a Noetherian domain with fraction field $K\ne R$. Let $\overline R$ be the integral closure of $R$ in $K$. Let us call that a subring of the fraction field $R\subseteq R^{fn}\subseteq K$ ...
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### In prime characteristic, is being $N$-1 a local property?

A Noetherian domain $R$ with fraction field $K$ is said to satisfy $N$-1 if the integral closure of $R$ in $K$ is module finite over $R$. My question is: Let $R$ be a Noetherian domain of finite ...
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### Polynomial ring over an analytically unramified ring is locally analytically unramified

Let us call a Noetherian local ring $(R, \mathfrak m)$ to be analytically unramified if the $\mathfrak m$-adic completion of $R$ is reduced i.e. has no non-zero nilpotent. https://en.m.wikipedia.org/...
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### Extending “the ring of exponents” from $\Bbb{Z}$ to that which contains a solution to $4^x - 3 = 0 \pmod 5$?

Let $R = \Bbb{Z}/5\Bbb{Z}$ be the ring of integers modulo $5$. Then $4^x = 1, 4, 1, 4, 1, 4, \dots$ as $x$ ranges over $\Bbb{Z}$. Thus $4^x = 3$ and $4^x = 2$ have no solution $x \in \Bbb{Z}$. ...
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### An equation defined by norm

Let $f$ be an Eisenstein polynomial of degree $n$ and the prime $p$. $\alpha$ is a root of $f$. Let $\mathbb{Q}(\alpha)=K$, Prove that for any $\gamma\in O_K$, there exist $a\in \mathbb{Z}$, such that ...
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### Example of integral extension of degree $p$?

For a prime $p$, does there exist an example of a (pair of) domains (if yes any way to construct?) $R$ and $S$ such that Both $R$ and $S$ are normal complete local domains $S$ contains an alg. ...
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### $K[x+y] \subseteq K[x, y]/(xy-1)$ is an integral extension

Let $K$ be a field. Prove that $K[x+y] \subseteq K[x, y]/(xy-1)$ is an integral extension. I know that $K[x,y]/(xy-1) \simeq K[t, t^{-1}]$, but I'm not sure if this would be useful to prove the ...
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### If $Q \in\mathrm{Spec}(B)$ is the unique prime ideal lying over $P \in\mathrm{Spec}(A)$, then $B_P=B_Q$.

Let $A \subseteq B$ be an integral extension of integral domains. Suppose $P \in \mathrm{Spec}(A)$ and $Q \in \mathrm{Spec}(B)$ is the unique prime ideal lying above $P$. Prove that $B_Q=B_P$. My ...
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### For an integral extension $A \subseteq B$, $\sqrt{IB}\cap A=\sqrt{I}$

Let $A \subseteq B$ be an integral extension and $I \subseteq A$ an ideal. Prove that $\sqrt{IB}\cap A=\sqrt{I}$. I saw a similar property holds for Jacobson radical where the following two facts ...
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### Integral extension of local rings of projective varieties

Let $X$ and $Y$ be projective varieties and $φ\colon X \to Y$ a regular birational map. If $q\in Y$ and $φ^{-1}(q)$ is finite and $φ(p) = q$, is it true that $\mathcal O_{X,p}$ is an integral ...
### $A \subset B$ be integral and flat extension then it is faithfully flat.
I want to prove the following: $A \subset B$ be integral and flat extension of rings then it is faithfully flat. Clearly enough to show that for every ideal $I$ of $A$, $I^{ec}=I$. Since the ...