Show that $[K(X):K(X^{p}, Y^{p})]=p^2$ and $[K(X):K(X^{p}, Y^{p})]_{i}=1$












0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    $[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


















0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    $[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
















0












0








0





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 16 '18 at 16:26







Manwell

















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




















  • $begingroup$
    $[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












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