Prove that C[x, y]/<xn + yn − 1> is an integral domain.

Gauss's lemma: A non-constant polynomial in ℤx is irreducible in ℤx if and only if it is both…

Answered: Prove that C[x, y]/ is an integral…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Prove that C[x, y]/<xⁿ + yⁿ − 1> is an integral domain.

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Step 1

Gauss's lemma: A non-constant polynomial in $ℤ [x]$ is irreducible in $ℤ [x]$ if and only if it is both irreducible in $ℚ [x]$ and primitive in $ℤ [x]$ .

Eisenstein criterion: Suppose that $f (x) \in ℤ [x]$ is given by

$f (x) = c_{n} x^{n} + . . . + c_{1} x + c_{0}$

(1) $p$ does not divide $c_{n}$

(2) $p$ divides $c_{n - 1}, . . ., p_{1}, p_{0}$

(3) $p^{2}$ does not divide $c_{0}$

Then, $f (x)$ is irreducible in $ℚ [x]$

$\frac{C [x, y]}{<x^{n} + y^{n} - 1>}$ is an integral domain if and only if $<x^{n} + y^{n} - 1>$ is prime in $C [x, y]$ if and only if $x^{n} + y^{n} - 1$ is irreducible in $C [x, y]$

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