Show that $[K(X):K(X^{p}, Y^{p})]=p^2$ and $[K(X):K(X^{p}, Y^{p})]_{i}=1$
$begingroup$
I don't know how to prove the following:
Let $K$ be a field. Then we have the polynomial ring $K[X_{1}, ..., X_{n}]$ and the field of fractions $$K(X_{1}, ..., X_{n}):= {f/g| f,g in K[X_{1}, ..., X_{n}], gneq 0}setminus sim $$
$f_{1}/g_{1} sim f_{2}/g_{2}$ means that $f_{1}g_{2}=f_{2}g_{1}$ in $K[X_{1}, ..., X_{n}]$.
i) Assume that $char(K)=p>0$ and $K subset K(X^{p}) subset K(X)$. Show that $[K(X):K(X^{p})]=p=[K(X):K(X^{p})]_{i}$.
ii) Assume that $char(K)=p>0$ and $K subset K(X^{p}, Y^{p}) subset K(X,Y)$. Use i) to show that $[K(X):K(X^{p}, Y^{p})]=p^2$ and $[K(X):K(X^{p}, Y^{p})]_{i}=1$.
For a finite field extension $Ksubset L$ is $ [L:K]_{i}$ defined as $[L:K]/[L:K]_{s} $ which is $1$ for $char(K)=0$ and $p^{e}$ for $char(K)=p>0, ein mathbb{N} $.
Thanks in advance for any help.
abstract-algebra
asked Dec 16 '18 at 0:49
ManwellManwell
114
$endgroup$
add a comment |
$begingroup$
I don't know how to prove the following:
Let $K$ be a field. Then we have the polynomial ring $K[X_{1}, ..., X_{n}]$ and the field of fractions $$K(X_{1}, ..., X_{n}):= {f/g| f,g in K[X_{1}, ..., X_{n}], gneq 0}setminus sim $$
$f_{1}/g_{1} sim f_{2}/g_{2}$ means that $f_{1}g_{2}=f_{2}g_{1}$ in $K[X_{1}, ..., X_{n}]$.
i) Assume that $char(K)=p>0$ and $K subset K(X^{p}) subset K(X)$. Show that $[K(X):K(X^{p})]=p=[K(X):K(X^{p})]_{i}$.
ii) Assume that $char(K)=p>0$ and $K subset K(X^{p}, Y^{p}) subset K(X,Y)$. Use i) to show that $[K(X):K(X^{p}, Y^{p})]=p^2$ and $[K(X):K(X^{p}, Y^{p})]_{i}=1$.
For a finite field extension $Ksubset L$ is $ [L:K]_{i}$ defined as $[L:K]/[L:K]_{s} $ which is $1$ for $char(K)=0$ and $p^{e}$ for $char(K)=p>0, ein mathbb{N} $.
Thanks in advance for any help.
abstract-algebra
asked Dec 16 '18 at 0:49
ManwellManwell
114
$endgroup$
$begingroup$
$[K(X,Y):K(X^p,Y^p)] = $ ? Do you know anything about the degree of inseparable extensions ? Does $K(X,Y)/K(X^p,Y^p)$ have a separable subextension ?
$endgroup$
– reuns
Dec 16 '18 at 2:05
$begingroup$
We just started with this topic in the lecture and I can't really grasp the concept yet to be honest. We had a few theorems but I don't see how they are supposed to help me.
$endgroup$
– Manwell
Dec 16 '18 at 3:20
$begingroup$
Do you see how for any field $F$ then $F(X), F(X^p)$ are fields and $F(X)$ is a $p$-dimensional $F(X^p)$ vector space so $[F(X):F(X^p)] = p$ ? So you get that $[K(X,Y):K(X^p:Y^p)] $ $= [K(X,Y):K(X,Y^p)][K(X,Y^p):K(X^p:Y^p)]$ $=[K(X)(Y):K(X)(Y^p)][K(Y^p)(X)K(Y^p)(X^p)] = p^2$. Then look at the minimal polynomials of those latter extensions to show (if $char(K) = p$) they are purely inseparable. Finally think about towers of purely inseparable extensions.
$endgroup$
– reuns
Dec 16 '18 at 3:25
add a comment |
$begingroup$
I don't know how to prove the following:
Let $K$ be a field. Then we have the polynomial ring $K[X_{1}, ..., X_{n}]$ and the field of fractions $$K(X_{1}, ..., X_{n}):= {f/g| f,g in K[X_{1}, ..., X_{n}], gneq 0}setminus sim $$
$f_{1}/g_{1} sim f_{2}/g_{2}$ means that $f_{1}g_{2}=f_{2}g_{1}$ in $K[X_{1}, ..., X_{n}]$.
i) Assume that $char(K)=p>0$ and $K subset K(X^{p}) subset K(X)$. Show that $[K(X):K(X^{p})]=p=[K(X):K(X^{p})]_{i}$.
ii) Assume that $char(K)=p>0$ and $K subset K(X^{p}, Y^{p}) subset K(X,Y)$. Use i) to show that $[K(X):K(X^{p}, Y^{p})]=p^2$ and $[K(X):K(X^{p}, Y^{p})]_{i}=1$.
For a finite field extension $Ksubset L$ is $ [L:K]_{i}$ defined as $[L:K]/[L:K]_{s} $ which is $1$ for $char(K)=0$ and $p^{e}$ for $char(K)=p>0, ein mathbb{N} $.
Thanks in advance for any help.
abstract-algebra
asked Dec 16 '18 at 0:49
ManwellManwell
114
$endgroup$
I don't know how to prove the following:
Let $K$ be a field. Then we have the polynomial ring $K[X_{1}, ..., X_{n}]$ and the field of fractions $$K(X_{1}, ..., X_{n}):= {f/g| f,g in K[X_{1}, ..., X_{n}], gneq 0}setminus sim $$
$f_{1}/g_{1} sim f_{2}/g_{2}$ means that $f_{1}g_{2}=f_{2}g_{1}$ in $K[X_{1}, ..., X_{n}]$.
i) Assume that $char(K)=p>0$ and $K subset K(X^{p}) subset K(X)$. Show that $[K(X):K(X^{p})]=p=[K(X):K(X^{p})]_{i}$.
ii) Assume that $char(K)=p>0$ and $K subset K(X^{p}, Y^{p}) subset K(X,Y)$. Use i) to show that $[K(X):K(X^{p}, Y^{p})]=p^2$ and $[K(X):K(X^{p}, Y^{p})]_{i}=1$.
For a finite field extension $Ksubset L$ is $ [L:K]_{i}$ defined as $[L:K]/[L:K]_{s} $ which is $1$ for $char(K)=0$ and $p^{e}$ for $char(K)=p>0, ein mathbb{N} $.
Thanks in advance for any help.
abstract-algebra
abstract-algebra
asked Dec 16 '18 at 0:49
ManwellManwell
114
asked Dec 16 '18 at 0:49
ManwellManwell
114
edited Dec 16 '18 at 16:26
Manwell
asked Dec 16 '18 at 0:49
ManwellManwell
114
asked Dec 16 '18 at 0:49
ManwellManwell
114
asked Dec 16 '18 at 0:49
ManwellManwell
114
114
$begingroup$
$[K(X,Y):K(X^p,Y^p)] = $ ? Do you know anything about the degree of inseparable extensions ? Does $K(X,Y)/K(X^p,Y^p)$ have a separable subextension ?
$endgroup$
– reuns
Dec 16 '18 at 2:05
$begingroup$
We just started with this topic in the lecture and I can't really grasp the concept yet to be honest. We had a few theorems but I don't see how they are supposed to help me.
$endgroup$
– Manwell
Dec 16 '18 at 3:20
$begingroup$
Do you see how for any field $F$ then $F(X), F(X^p)$ are fields and $F(X)$ is a $p$-dimensional $F(X^p)$ vector space so $[F(X):F(X^p)] = p$ ? So you get that $[K(X,Y):K(X^p:Y^p)] $ $= [K(X,Y):K(X,Y^p)][K(X,Y^p):K(X^p:Y^p)]$ $=[K(X)(Y):K(X)(Y^p)][K(Y^p)(X)K(Y^p)(X^p)] = p^2$. Then look at the minimal polynomials of those latter extensions to show (if $char(K) = p$) they are purely inseparable. Finally think about towers of purely inseparable extensions.
$endgroup$
– reuns
Dec 16 '18 at 3:25
add a comment |
$begingroup$
$[K(X,Y):K(X^p,Y^p)] = $ ? Do you know anything about the degree of inseparable extensions ? Does $K(X,Y)/K(X^p,Y^p)$ have a separable subextension ?
$endgroup$
– reuns
Dec 16 '18 at 2:05
$begingroup$
We just started with this topic in the lecture and I can't really grasp the concept yet to be honest. We had a few theorems but I don't see how they are supposed to help me.
$endgroup$
– Manwell
Dec 16 '18 at 3:20
$begingroup$
Do you see how for any field $F$ then $F(X), F(X^p)$ are fields and $F(X)$ is a $p$-dimensional $F(X^p)$ vector space so $[F(X):F(X^p)] = p$ ? So you get that $[K(X,Y):K(X^p:Y^p)] $ $= [K(X,Y):K(X,Y^p)][K(X,Y^p):K(X^p:Y^p)]$ $=[K(X)(Y):K(X)(Y^p)][K(Y^p)(X)K(Y^p)(X^p)] = p^2$. Then look at the minimal polynomials of those latter extensions to show (if $char(K) = p$) they are purely inseparable. Finally think about towers of purely inseparable extensions.
$endgroup$
– reuns
Dec 16 '18 at 3:25
$begingroup$
$[K(X,Y):K(X^p,Y^p)] = $ ? Do you know anything about the degree of inseparable extensions ? Does $K(X,Y)/K(X^p,Y^p)$ have a separable subextension ?
$endgroup$
– reuns
Dec 16 '18 at 2:05
$begingroup$
$[K(X,Y):K(X^p,Y^p)] = $ ? Do you know anything about the degree of inseparable extensions ? Does $K(X,Y)/K(X^p,Y^p)$ have a separable subextension ?
$endgroup$
– reuns
Dec 16 '18 at 2:05
$begingroup$
We just started with this topic in the lecture and I can't really grasp the concept yet to be honest. We had a few theorems but I don't see how they are supposed to help me.
$endgroup$
– Manwell
Dec 16 '18 at 3:20
$begingroup$
We just started with this topic in the lecture and I can't really grasp the concept yet to be honest. We had a few theorems but I don't see how they are supposed to help me.
$endgroup$
– Manwell
Dec 16 '18 at 3:20
$begingroup$
Do you see how for any field $F$ then $F(X), F(X^p)$ are fields and $F(X)$ is a $p$-dimensional $F(X^p)$ vector space so $[F(X):F(X^p)] = p$ ? So you get that $[K(X,Y):K(X^p:Y^p)] $ $= [K(X,Y):K(X,Y^p)][K(X,Y^p):K(X^p:Y^p)]$ $=[K(X)(Y):K(X)(Y^p)][K(Y^p)(X)K(Y^p)(X^p)] = p^2$. Then look at the minimal polynomials of those latter extensions to show (if $char(K) = p$) they are purely inseparable. Finally think about towers of purely inseparable extensions.
$endgroup$
– reuns
Dec 16 '18 at 3:25
$begingroup$
Do you see how for any field $F$ then $F(X), F(X^p)$ are fields and $F(X)$ is a $p$-dimensional $F(X^p)$ vector space so $[F(X):F(X^p)] = p$ ? So you get that $[K(X,Y):K(X^p:Y^p)] $ $= [K(X,Y):K(X,Y^p)][K(X,Y^p):K(X^p:Y^p)]$ $=[K(X)(Y):K(X)(Y^p)][K(Y^p)(X)K(Y^p)(X^p)] = p^2$. Then look at the minimal polynomials of those latter extensions to show (if $char(K) = p$) they are purely inseparable. Finally think about towers of purely inseparable extensions.
$endgroup$
– reuns
Dec 16 '18 at 3:25
add a comment |
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$begingroup$
$[K(X,Y):K(X^p,Y^p)] = $ ? Do you know anything about the degree of inseparable extensions ? Does $K(X,Y)/K(X^p,Y^p)$ have a separable subextension ?
$endgroup$
– reuns
Dec 16 '18 at 2:05
$begingroup$
We just started with this topic in the lecture and I can't really grasp the concept yet to be honest. We had a few theorems but I don't see how they are supposed to help me.
$endgroup$
– Manwell
Dec 16 '18 at 3:20
$begingroup$
Do you see how for any field $F$ then $F(X), F(X^p)$ are fields and $F(X)$ is a $p$-dimensional $F(X^p)$ vector space so $[F(X):F(X^p)] = p$ ? So you get that $[K(X,Y):K(X^p:Y^p)] $ $= [K(X,Y):K(X,Y^p)][K(X,Y^p):K(X^p:Y^p)]$ $=[K(X)(Y):K(X)(Y^p)][K(Y^p)(X)K(Y^p)(X^p)] = p^2$. Then look at the minimal polynomials of those latter extensions to show (if $char(K) = p$) they are purely inseparable. Finally think about towers of purely inseparable extensions.
$endgroup$
– reuns
Dec 16 '18 at 3:25