Irreducibility of $x_1^2+x_2^2+...+x_n^2$ in $\mathbb{C}[x_1,...,x_n]$

Asked Nov 03 '21 at 15:13

Active Nov 03 '21 at 15:26

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I wanted to know whether the polynomial function $x_1^2+...+x_n^2 \in \mathbb{C}[x_1,...,x_n]$ is irreducible in $\mathbb{C}[x_1,...,x_n]$ for $3 \leq n$ ?

edited Nov 03 '21 at 15:26

asked Nov 03 '21 at 15:13

Koalalover

2

First thing that comes to mind is induction with Eisenstein's? I haven't checked this. – Travis62 Nov 03 '21 at 15:15
ok: correct me, if i`m wrong: so if i know that $x_1^2+...+x_{n-1}^2$ is irreducible and therefore prime. Then $f_n=f_{n-1}+x_n^2$ is irreducible because i can use $f_{n-1}$ as the prime element which is needed for the Eisenstein criterion? – Koalalover Nov 03 '21 at 16:28

Irreducibility of $x_1^2+x_2^2+...+x_n^2$ in $\mathbb{C}[x_1,...,x_n]$

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