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 Post subject: Irreducibility of plane curvesPosted: Fri, 27 Apr 2012 19:27:08 UTC
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So I was looking through past exam questions with a friend and we came across this one:

"Show that is irreducible over ."

My first thought was that it's obviously true because because I can't factor anything out, but I have no idea how to prove it rigorously enough for exam-standard.

I thought something along the lines of
Assuming it can be written as the product of a line and a lower dimensional curve. Then

Would I also need to consider the case where I can pull out some sort of conic as well?

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 Post subject: Re: Irreducibility of plane curvesPosted: Fri, 27 Apr 2012 19:38:41 UTC
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peccavi_2006 wrote:
So I was looking through past exam questions with a friend and we came across this one:

"Show that is irreducible over ."

My first thought was that it's obviously true because because I can't factor anything out, but I have no idea how to prove it rigorously enough for exam-standard.

I thought something along the lines of
Assuming it can be written as the product of a line and a lower dimensional curve. Then

Would I also need to consider the case where I can pull out some sort of conic as well?

You need more than just .

Recall is a UFD.

Suppose be a (nontrivial) factorisation. Then taking degrees in y, you have . So we may assume WLOG (since the presence of the y^3 term means you cannot have --- in exam you need to expand that a little bit more).

Hence we may assume , where p(x) divides x^4, so is one of . Now examine each one in turn.

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 Post subject: Re: Irreducibility of plane curvesPosted: Fri, 27 Apr 2012 19:44:18 UTC
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I assume because we can't have that , right? Hence there must be some g component in there.

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 Post subject: Re: Irreducibility of plane curvesPosted: Fri, 27 Apr 2012 19:57:49 UTC
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peccavi_2006 wrote:
I assume because we can't have that , right? Hence there must be some g component in there.

No. means g is a polynomial in x alone. Thus when we write f as a polynomial in y with coefficients in , every coefficient must be divisible by g. That means (1=coefficient of y^3), which gives trivial factorisation.

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 Post subject: Re: Irreducibility of plane curvesPosted: Fri, 27 Apr 2012 20:28:07 UTC
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Posts: 650
Location: England
ofc yeah - thanks outermeasure.

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