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Questions tagged [integral-extensions]

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

Let $~A ⊆ B~$ be a ring extension. Generalizing to rings the notion of algebraic elements and extensions from field theory, we say that an element $~b~$ in $~B~$ is integral over $~A~$ if there exists a monic polynomial $~f(x)~$ in the variable $~x~$ with coefficients in $~A~$ such that $~f(b) = 0~$; we say that the extension is integral if every element of $~B~$ is integral over $~A~$.

  • A domain that is an integral extension of a field must be a field

Reference:

https://en.wikipedia.org/wiki/Integral_element#Integral_extensions

"Integral Extensions" by Siegfried Bosch

179 questions
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If a polynomial in $B[x]$ is integral over $A[x]$, then are its coefficients integral over $A$?

Suppose $A \subset B$ are integral domains, and $f = b_0 + b_1 x + \ldots + b_m x^m$, where $b_k \in B$. Is it true that if $f$ is integral over $A[x]$ then all the coefficients $b_k$ are integral over $A$? Note that the other way is obvious, i.e.…
Rodion Zaytsev
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How can I compute the discriminant of the field $\mathbb{Q}(\sqrt[3]{28})$?

$$\newcommand{\Q}{\mathbb{Q}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\al}{\alpha} \newcommand{\bcal}{\mathcal{B}} \newcommand{\qroot}{\sqrt[3]} \newcommand{\froot}{\sqrt[4]} $$ I have a problem which consist in 3…
Dr Richard Clare
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Integral domain over which every non-constant irreducible polynomial has degree 1

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Non-flatness of $k[t]$ as a $k[t^2,t^3]$-module

Let $k$ be a field. How to show that $k[t]$ is not flat as a module over $k[t^2,t^3]$ ? Since the ring extension $k[t^2,t^3]\subseteq k[t]$ is integral, it is clear that $k[t]$ is a finitely generated $k[t^2,t^3]$-module , and also torsion free. I…
user
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The Krull dimension of a ring R is defined as the supremum of the lengths of chains of prime ideals contained in R. I heard that an integral extension over a ring R has the same Krull dimension as R, however, I don't really see why this is true.
McNuggets666
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FTem
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Ángel Valencia
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For a recursion based on $x+y+z+2\sqrt{xy+yz+zx},$ does what happens in $\mathbb Z\left[\sqrt n\right]$ stay in $\mathbb Z\left[\sqrt n\right]$?

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There are various approaches how to generalize the notion of an algebraically closed field to the context of commutative rings. A good survey is R. Raphael, On algebraic closures. I am interested in the following notion of an "algebraically closed"…
Martin Brandenburg
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In an integral extension $A\subseteq B$, when $B$ Noetherian implies $A$ Noetherian?

Let $A\subseteq B$ be an extension of commutative rings. If $B$ is a Noetherian ring and finitely generated as $A$-module, then $A$ is Noetherian ring. Is there any other such criteria, under possibly some additional conditions, which says that in…
user495643
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Is there an integral extension of $\mathbb{Z}$ keeping its prime elements?

Let $R$ be a ring. An extension $S$ of $R$ is called prime-keeping if every element $p$ prime in $R$ is also prime in $S$. Consider the ring $\mathbb{Z}$ and the following two extensions: $\mathbb{Z}[X]$ and $\mathbb{Z}[i]$. The first one is…
Sebastien Palcoux
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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 $S$, and that if $R$ is integrally closed then so is…
user302934
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Is $\mathbb{Z}[x,y]/(x^3+px+q - y^2)$ integral over $\mathbb{Z}[x]$?

Let $ B = \mathbb{Z}[x,y]/(x^3+px+q - y^2)$ and $A = \mathbb{Z}[x]$. I want to know whether the ring extension $A \subset B$ is integral or not. There can be two possibilities: $x^3+px+q$ is irreducible or is not irreducible. It's enough to…
Lada Dudnikova
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Alphonse
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