Mixing
Galois extension - minimum polynomial
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Let $K$ be a Galois extension of $F$ and let $ain K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=text{Gal}(K/F)$ and $H=text{Gal}(K/F(a))$.
We symbolize with $tau_1, ldots , tau_r$ the left cosets of $H$ in $G$.
Show that $displaystyle{min (F,a)=prod_{i=1}^rleft (x-tau_i(a)right )}$.
Show that $displaystyle{prod_{sigma in G}left (x-tau_i(a)right )=min (F,a)^{n/r}}$.
$$$$
Could you give me a hint how we could show these two points? I don't really have an idea.
abstract-algebra field-theory galois-theory
asked 2 days ago
Mary Star
2,88582263
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up vote
-1
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favorite
Let $K$ be a Galois extension of $F$ and let $ain K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=text{Gal}(K/F)$ and $H=text{Gal}(K/F(a))$.
We symbolize with $tau_1, ldots , tau_r$ the left cosets of $H$ in $G$.
Show that $displaystyle{min (F,a)=prod_{i=1}^rleft (x-tau_i(a)right )}$.
Show that $displaystyle{prod_{sigma in G}left (x-tau_i(a)right )=min (F,a)^{n/r}}$.
$$$$
Could you give me a hint how we could show these two points? I don't really have an idea.
abstract-algebra field-theory galois-theory
asked 2 days ago
Mary Star
2,88582263
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let $K$ be a Galois extension of $F$ and let $ain K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=text{Gal}(K/F)$ and $H=text{Gal}(K/F(a))$.
We symbolize with $tau_1, ldots , tau_r$ the left cosets of $H$ in $G$.
Show that $displaystyle{min (F,a)=prod_{i=1}^rleft (x-tau_i(a)right )}$.
Show that $displaystyle{prod_{sigma in G}left (x-tau_i(a)right )=min (F,a)^{n/r}}$.
$$$$
Could you give me a hint how we could show these two points? I don't really have an idea.
abstract-algebra field-theory galois-theory
asked 2 days ago
Mary Star
2,88582263
Let $K$ be a Galois extension of $F$ and let $ain K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=text{Gal}(K/F)$ and $H=text{Gal}(K/F(a))$.
We symbolize with $tau_1, ldots , tau_r$ the left cosets of $H$ in $G$.
Show that $displaystyle{min (F,a)=prod_{i=1}^rleft (x-tau_i(a)right )}$.
Show that $displaystyle{prod_{sigma in G}left (x-tau_i(a)right )=min (F,a)^{n/r}}$.
$$$$
Could you give me a hint how we could show these two points? I don't really have an idea.
abstract-algebra field-theory galois-theory
abstract-algebra field-theory galois-theory
asked 2 days ago
Mary Star
2,88582263
asked 2 days ago
Mary Star
2,88582263
asked 2 days ago
Mary Star
2,88582263
asked 2 days ago
Mary Star
2,88582263
asked 2 days ago
Mary Star
2,88582263
2,88582263
add a comment |
add a comment |
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