Example of polynomial in two variables
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Can you please give me an example of a polynomial $F in K[X,Y]$ such that $V(F)$ is finite?
I found in Fulton the following proposition:
If F is an irreducible polynomial in $K[X,Y]$ such that $V(F)$ is infinite, then $I(V(F))=(F)$, and $V(F)$ is irreducible.
Is the condition $V(F)$ is infinite necessarily?
Thank you!
algebraic-geometry polynomials irreducible-polynomials
asked Nov 17 at 7:52
mip
344
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up vote
0
down vote
favorite
Can you please give me an example of a polynomial $F in K[X,Y]$ such that $V(F)$ is finite?
I found in Fulton the following proposition:
If F is an irreducible polynomial in $K[X,Y]$ such that $V(F)$ is infinite, then $I(V(F))=(F)$, and $V(F)$ is irreducible.
Is the condition $V(F)$ is infinite necessarily?
Thank you!
algebraic-geometry polynomials irreducible-polynomials
asked Nov 17 at 7:52
mip
344
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Can you please give me an example of a polynomial $F in K[X,Y]$ such that $V(F)$ is finite?
I found in Fulton the following proposition:
If F is an irreducible polynomial in $K[X,Y]$ such that $V(F)$ is infinite, then $I(V(F))=(F)$, and $V(F)$ is irreducible.
Is the condition $V(F)$ is infinite necessarily?
Thank you!
algebraic-geometry polynomials irreducible-polynomials
asked Nov 17 at 7:52
mip
344
Can you please give me an example of a polynomial $F in K[X,Y]$ such that $V(F)$ is finite?
I found in Fulton the following proposition:
If F is an irreducible polynomial in $K[X,Y]$ such that $V(F)$ is infinite, then $I(V(F))=(F)$, and $V(F)$ is irreducible.
Is the condition $V(F)$ is infinite necessarily?
Thank you!
algebraic-geometry polynomials irreducible-polynomials
algebraic-geometry polynomials irreducible-polynomials
asked Nov 17 at 7:52
mip
344
asked Nov 17 at 7:52
mip
344
asked Nov 17 at 7:52
mip
344
asked Nov 17 at 7:52
mip
344
asked Nov 17 at 7:52
mip
344
344
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
Take $K = mathbb{R}$ and $F(X,Y) = X^2 + Y^2$.
answered Nov 17 at 7:56
Sasha
4,09339
add a comment |
up vote
2
down vote
Here's a very easy way: take $K$ to be a finite field and $F$ to be any polynomial in $K[X,Y]$.
answered Nov 17 at 17:01
André 3000
12.2k22041
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Take $K = mathbb{R}$ and $F(X,Y) = X^2 + Y^2$.
answered Nov 17 at 7:56
Sasha
4,09339
add a comment |
up vote
2
down vote
accepted
Take $K = mathbb{R}$ and $F(X,Y) = X^2 + Y^2$.
answered Nov 17 at 7:56
Sasha
4,09339
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Take $K = mathbb{R}$ and $F(X,Y) = X^2 + Y^2$.
answered Nov 17 at 7:56
Sasha
4,09339
Take $K = mathbb{R}$ and $F(X,Y) = X^2 + Y^2$.
answered Nov 17 at 7:56
Sasha
4,09339
answered Nov 17 at 7:56
Sasha
4,09339
answered Nov 17 at 7:56
Sasha
4,09339
answered Nov 17 at 7:56
Sasha
4,09339
4,09339
add a comment |
add a comment |
up vote
2
down vote
Here's a very easy way: take $K$ to be a finite field and $F$ to be any polynomial in $K[X,Y]$.
answered Nov 17 at 17:01
André 3000
12.2k22041
add a comment |
up vote
2
down vote
Here's a very easy way: take $K$ to be a finite field and $F$ to be any polynomial in $K[X,Y]$.
answered Nov 17 at 17:01
André 3000
12.2k22041
add a comment |
up vote
2
down vote
up vote
2
down vote
Here's a very easy way: take $K$ to be a finite field and $F$ to be any polynomial in $K[X,Y]$.
answered Nov 17 at 17:01
André 3000
12.2k22041
Here's a very easy way: take $K$ to be a finite field and $F$ to be any polynomial in $K[X,Y]$.
answered Nov 17 at 17:01
André 3000
12.2k22041
answered Nov 17 at 17:01
André 3000
12.2k22041
answered Nov 17 at 17:01
André 3000
12.2k22041
answered Nov 17 at 17:01
André 3000
12.2k22041
12.2k22041
add a comment |
add a comment |
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