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I was doing Ravi's FOAG exercise, in 5.4I (see related questionhere) there is a question ask me to show that

$$\operatorname{Spec}\left(k[x_1, \ldots, x_n]/(x_1^2 + \cdots + x_m^2)\right) $$

is normal for $n\ge m\ge 3$ and $\text{char} k \ne 2$.

The idea is to use the previous result in 5.4H , therefore only need to check the question here satisfies the condition.therefore the problem is reduced to show that:

$x_1^2+\cdots+x_m^2 \in k[x_1,\ldots,x_n]$ is irreducible for $n\ge m \ge 3$, and $x_1^2+x_2^2 \in k[x_1,\ldots,x_n]$ is either irreducible or do not has repeated prime factor.

how to prove the irreducibility of it?

joriki
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yi li
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    Perhaps this is a bit "overkill", but $x_1^2 + \dots + x_m^2$ is a quadric in $\mathbf{P}^n$ and it is non-singular. I think that a reducible hypersurface in $\mathbf{P}^n$ should have a singular point. Usually you argue that two of its components must intersect somewhere and for this you use resultants. I will try later if I have time – Joseph Harrison Jan 18 '23 at 13:12
  • thank you Joseph , I found a solution in the case of $\Bbb{C}$ which has many different approaches. here https://math.stackexchange.com/q/486668/360262 – yi li Jan 18 '23 at 13:13
  • I am reading Vakil's book as well, although I haven't got as far as you have yet – Joseph Harrison Jan 18 '23 at 13:15
  • Vakil's FOAG is very good place to learn AG, since it contains a lot of exercise and concrete example to compute – yi li Jan 18 '23 at 13:19
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    I completely agree. Before Vakil's book I was using Hartshorne's, which can be quite unintuitive. Also my first comment about the quadric in $\mathbf{P}^n$ is wrong: there can be many singular points if $n > m$, for example take $x_1 = \dots = x_m = 0$ and $x_{m + 1} = \dots = x_n = 1$. I will delete it once you have seen this – Joseph Harrison Jan 18 '23 at 16:56
  • Yeah, thank you Joseph. I heard someone says that the most interesting part of Hartshorne is the exercise. – yi li Jan 19 '23 at 02:13
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    See also this notes: https://www-users.cse.umn.edu/~garrett/m/algebra/notes/16.pdf – yi li Jan 21 '23 at 08:37
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    that is a very good solution: use Eisenstein's criterion with the UFD $k[x_2, \dots, x_m]$ to see inductively that $x_1^2 + \dots + x_m^2$ is irreducible. – Joseph Harrison Jan 21 '23 at 14:50

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