Let A be a discrete valuation ring, and let&nbsp;a∈A&nbsp;be a non-zero element. Compute the integral closure of&nbsp;B:=A[X,&nbsp;aX]

Step 1You have to show something more: the field of fractions of&nbsp;B&nbsp;equals the field of fractions ofA[X,&nbsp;uX]&nbsp;for&nbsp;n=2m, and similarly for n odd.But this is easy enough since&nbsp;uX=αmuXαm∈Q(B)(Q(B)&nbsp;denotes the field of fractions of&nbsp;B.)

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Papierskiix5n

Answered question

2022-04-15

Let A be a discrete valuation ring, and let $a \in A$ be a non-zero element. Compute the integral closure of $B : = A [X, \sqrt{a X}]$

Answer & Explanation

Marin Lowe

Beginner2022-04-16Added 18 answers

Step 1
You have to show something more: the field of fractions of $B$ equals the field of fractions of
$A [X, \sqrt{u X}]$ for $n = 2 m$ , and similarly for n odd.
But this is easy enough since $\sqrt{u X} = \frac{α^{m} \sqrt{u X}}{α^{m}}$ $\in Q (B)$
( $Q (B)$ denotes the field of fractions of $B$ .)

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