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Can any one help me, how to show that a hypersurface in $\mathbb{A}^n$ is irreducible iff the defining equation F is a power of an irreducible polynomial G(i.e G can not be written as a product of two non constant polynomial). thank you. here I want to inform that I don't know what is irreducible hypersurface.

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start here: – Blah Apr 5 '12 at 17:46
Blah's link says "An algebraic variety is called irreducible if it cannot be written as the union of nonempty algebraic varieties" . That definition is hilariously false. Here is a correct definition : a variety $X$ is irreducible if it cannot be written as the union of two closed subvarieties $Y,Y'\subsetneq X$ strictly included in the given variety $X$. – Georges Elencwajg Apr 5 '12 at 21:41
I find it quite extraordinary that you do not ask what an irreducible hypersurface is... It should be obvious to you that it is simply impossible to prove what you want without knowing what an irreducible hypersurface is! Out of curiosity: why are you trying to prove a characterization of a notion that you don't know? – Mariano Suárez-Alvarez Apr 5 '12 at 22:39
up vote 5 down vote accepted

For $k$ an algebraically closed field, a hypersurface $H\subset \mathbb A^n_k$ is a closed subset of the form $H=V(F)$, where $F\in k[T_1,...,T_n]\setminus k \;$ is some non-constant polynomial.
The easy irreducibility criterion for $H$ (like for any closed subset of $\mathbb A^n_k $) is that the ideal of polynomials vanishing on it, $I(H)\subset k[T_1,...,Tn]$, be prime.
Now, Hilbert's Nullstellensatz computes that radical for us:
$$I(H)=I(V(F))=rad (F)$$ If the factorization of $F\in k[T_1,...,T_n]$ into irreducible polynomials is $F=G_1^{n_1}\cdot G_2^{n_2} \cdot \ldots \cdot G_r^{n_r}$, we have $$ rad (F)= (G_1\cdot G_2 \cdot \ldots \cdot G_r ) $$ and this will be a prime ideal iff $r=1$ i.e. finally iff $F=G^r$ with $G$ an irreducible polynomial .

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This is an exercise in chapter 1 on page 12 in Smith's Invitation to Algebraic Geometry. At this point in the book ideals have not been mentioned. Neither has Hilbert's Nullstellensatz. Is there any way to answer this exercise only using the definitions or similarly minimal machinery? – student Mar 13 at 1:34

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