Galois extension - minimum polynomial
up vote
0
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 representatives of the left cosets of $H$ in $G$.
Show that $displaystyle{min (F,a)=prod_{i=1}^rleft (x-tau_i(a)right )}$.
Show that $displaystyle{prod_{sigma in G}left (x- tau_i(a)right )=min (F,a)^{n/r}}$, where $tau_i$ is the unique representative such that $sigma in left[ tau_i right]$.
$$$$
Could you give me a hint how we could show these two points? I don't really have an idea.
abstract-algebra field-theory galois-theory
edited Nov 29 at 14:46
Bilo
1089
asked Nov 19 at 23:09
Mary Star
2,92082266
add a comment |
up vote
0
down vote
favorite
Let $K$ be a Galois extension of $F$ and let $ain K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=text{Gal}(K/F)$ and $H=text{Gal}(K/F(a))$.
We symbolize with $tau_1, ldots , tau_r$ the representatives of the left cosets of $H$ in $G$.
Show that $displaystyle{min (F,a)=prod_{i=1}^rleft (x-tau_i(a)right )}$.
Show that $displaystyle{prod_{sigma in G}left (x- tau_i(a)right )=min (F,a)^{n/r}}$, where $tau_i$ is the unique representative such that $sigma in left[ tau_i right]$.
$$$$
Could you give me a hint how we could show these two points? I don't really have an idea.
abstract-algebra field-theory galois-theory
edited Nov 29 at 14:46
Bilo
1089
asked Nov 19 at 23:09
Mary Star
2,92082266
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $K$ be a Galois extension of $F$ and let $ain K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=text{Gal}(K/F)$ and $H=text{Gal}(K/F(a))$.
We symbolize with $tau_1, ldots , tau_r$ the representatives of the left cosets of $H$ in $G$.
Show that $displaystyle{min (F,a)=prod_{i=1}^rleft (x-tau_i(a)right )}$.
Show that $displaystyle{prod_{sigma in G}left (x- tau_i(a)right )=min (F,a)^{n/r}}$, where $tau_i$ is the unique representative such that $sigma in left[ tau_i right]$.
$$$$
Could you give me a hint how we could show these two points? I don't really have an idea.
abstract-algebra field-theory galois-theory
edited Nov 29 at 14:46
Bilo
1089
asked Nov 19 at 23:09
Mary Star
2,92082266
Let $K$ be a Galois extension of $F$ and let $ain K$. Let $n=[K:F]$, $r=[F(a):F]$, $G=text{Gal}(K/F)$ and $H=text{Gal}(K/F(a))$.
We symbolize with $tau_1, ldots , tau_r$ the representatives of the left cosets of $H$ in $G$.
Show that $displaystyle{min (F,a)=prod_{i=1}^rleft (x-tau_i(a)right )}$.
Show that $displaystyle{prod_{sigma in G}left (x- tau_i(a)right )=min (F,a)^{n/r}}$, where $tau_i$ is the unique representative such that $sigma in left[ tau_i right]$.
$$$$
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
edited Nov 29 at 14:46
Bilo
1089
asked Nov 19 at 23:09
Mary Star
2,92082266
edited Nov 29 at 14:46
Bilo
1089
asked Nov 19 at 23:09
Mary Star
2,92082266
edited Nov 29 at 14:46
Bilo
1089
edited Nov 29 at 14:46
Bilo
1089
edited Nov 29 at 14:46
Bilo
1089
1089
asked Nov 19 at 23:09
Mary Star
2,92082266
asked Nov 19 at 23:09
Mary Star
2,92082266
asked Nov 19 at 23:09
Mary Star
2,92082266
2,92082266
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
For the first point, try to show that if $i neq j$, then $tau_i(a) neq tau_j(a)$.
For the second one, try to show that if $sigma, tau$ belong to the same class modulo $H$, then $sigma(a) = tau(a)$
answered Nov 23 at 12:52
Bilo
1089
1
Do I have to write a full proof?
– Bilo
Nov 23 at 20:39
For the first part: Do they have to be different because the number of the left costes is equal to the extension degree $[F(a):F]$ ?
– Mary Star
Nov 24 at 8:38
1
Yes, since $tau_i$ maps a root of a polynomial into a root of the same polynomial. With this claim you can show that $tau_i(a)$ define all' the roots of $min(F,a)$, since it's degree is exactly $r$, so It must have the form you have described above
– Bilo
Nov 24 at 14:57
Ah ok!! Could you explain to me also further the second point?
– Mary Star
Nov 24 at 16:41
1
Yes, for sure. First show that if $sigma,tau$ belong to the same class modulo $H$, then $sigma(a)=tau(a)$; Since $vert G:H vert = r$, each coset contains exacly $frac{n}{r}$ elements. Denote with $C_1,...,C_r$ the distinct cosets of $G$ modulo $H$., then begin{align} prod_{sigma in G}(x-tau_i(a)) &= prod_{sigma in C_1}(x-tau_1(a)) cdots prod_{sigma in C_r}(x-tau_r(a)) \ &= (x-tau_1(a))^{n/r} cdots (x-tau_r(a))^{n/r} \ &= (prod_{1}^r(x-tau_i(a)) )^{n/r} \ &= min(F,a)^{n/r} end{align} Hope it's clear.
– Bilo
Nov 24 at 17:19
|
show 1 more comment
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005655%2fgalois-extension-minimum-polynomial%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
For the first point, try to show that if $i neq j$, then $tau_i(a) neq tau_j(a)$.
For the second one, try to show that if $sigma, tau$ belong to the same class modulo $H$, then $sigma(a) = tau(a)$
answered Nov 23 at 12:52
Bilo
1089
1
Do I have to write a full proof?
– Bilo
Nov 23 at 20:39
For the first part: Do they have to be different because the number of the left costes is equal to the extension degree $[F(a):F]$ ?
– Mary Star
Nov 24 at 8:38
1
Yes, since $tau_i$ maps a root of a polynomial into a root of the same polynomial. With this claim you can show that $tau_i(a)$ define all' the roots of $min(F,a)$, since it's degree is exactly $r$, so It must have the form you have described above
– Bilo
Nov 24 at 14:57
Ah ok!! Could you explain to me also further the second point?
– Mary Star
Nov 24 at 16:41
1
Yes, for sure. First show that if $sigma,tau$ belong to the same class modulo $H$, then $sigma(a)=tau(a)$; Since $vert G:H vert = r$, each coset contains exacly $frac{n}{r}$ elements. Denote with $C_1,...,C_r$ the distinct cosets of $G$ modulo $H$., then begin{align} prod_{sigma in G}(x-tau_i(a)) &= prod_{sigma in C_1}(x-tau_1(a)) cdots prod_{sigma in C_r}(x-tau_r(a)) \ &= (x-tau_1(a))^{n/r} cdots (x-tau_r(a))^{n/r} \ &= (prod_{1}^r(x-tau_i(a)) )^{n/r} \ &= min(F,a)^{n/r} end{align} Hope it's clear.
– Bilo
Nov 24 at 17:19
|
show 1 more comment
up vote
2
down vote
accepted
For the first point, try to show that if $i neq j$, then $tau_i(a) neq tau_j(a)$.
For the second one, try to show that if $sigma, tau$ belong to the same class modulo $H$, then $sigma(a) = tau(a)$
answered Nov 23 at 12:52
Bilo
1089
1
Do I have to write a full proof?
– Bilo
Nov 23 at 20:39
For the first part: Do they have to be different because the number of the left costes is equal to the extension degree $[F(a):F]$ ?
– Mary Star
Nov 24 at 8:38
1
Yes, since $tau_i$ maps a root of a polynomial into a root of the same polynomial. With this claim you can show that $tau_i(a)$ define all' the roots of $min(F,a)$, since it's degree is exactly $r$, so It must have the form you have described above
– Bilo
Nov 24 at 14:57
Ah ok!! Could you explain to me also further the second point?
– Mary Star
Nov 24 at 16:41
1
Yes, for sure. First show that if $sigma,tau$ belong to the same class modulo $H$, then $sigma(a)=tau(a)$; Since $vert G:H vert = r$, each coset contains exacly $frac{n}{r}$ elements. Denote with $C_1,...,C_r$ the distinct cosets of $G$ modulo $H$., then begin{align} prod_{sigma in G}(x-tau_i(a)) &= prod_{sigma in C_1}(x-tau_1(a)) cdots prod_{sigma in C_r}(x-tau_r(a)) \ &= (x-tau_1(a))^{n/r} cdots (x-tau_r(a))^{n/r} \ &= (prod_{1}^r(x-tau_i(a)) )^{n/r} \ &= min(F,a)^{n/r} end{align} Hope it's clear.
– Bilo
Nov 24 at 17:19
|
show 1 more comment
up vote
2
down vote
accepted
up vote
2
down vote
accepted
For the first point, try to show that if $i neq j$, then $tau_i(a) neq tau_j(a)$.
For the second one, try to show that if $sigma, tau$ belong to the same class modulo $H$, then $sigma(a) = tau(a)$
answered Nov 23 at 12:52
Bilo
1089
For the first point, try to show that if $i neq j$, then $tau_i(a) neq tau_j(a)$.
For the second one, try to show that if $sigma, tau$ belong to the same class modulo $H$, then $sigma(a) = tau(a)$
answered Nov 23 at 12:52
Bilo
1089
answered Nov 23 at 12:52
Bilo
1089
answered Nov 23 at 12:52
Bilo
1089
answered Nov 23 at 12:52
Bilo
1089
1089
1
Do I have to write a full proof?
– Bilo
Nov 23 at 20:39
For the first part: Do they have to be different because the number of the left costes is equal to the extension degree $[F(a):F]$ ?
– Mary Star
Nov 24 at 8:38
1
Yes, since $tau_i$ maps a root of a polynomial into a root of the same polynomial. With this claim you can show that $tau_i(a)$ define all' the roots of $min(F,a)$, since it's degree is exactly $r$, so It must have the form you have described above
– Bilo
Nov 24 at 14:57
Ah ok!! Could you explain to me also further the second point?
– Mary Star
Nov 24 at 16:41
1
Yes, for sure. First show that if $sigma,tau$ belong to the same class modulo $H$, then $sigma(a)=tau(a)$; Since $vert G:H vert = r$, each coset contains exacly $frac{n}{r}$ elements. Denote with $C_1,...,C_r$ the distinct cosets of $G$ modulo $H$., then begin{align} prod_{sigma in G}(x-tau_i(a)) &= prod_{sigma in C_1}(x-tau_1(a)) cdots prod_{sigma in C_r}(x-tau_r(a)) \ &= (x-tau_1(a))^{n/r} cdots (x-tau_r(a))^{n/r} \ &= (prod_{1}^r(x-tau_i(a)) )^{n/r} \ &= min(F,a)^{n/r} end{align} Hope it's clear.
– Bilo
Nov 24 at 17:19
|
show 1 more comment
1
Do I have to write a full proof?
– Bilo
Nov 23 at 20:39
For the first part: Do they have to be different because the number of the left costes is equal to the extension degree $[F(a):F]$ ?
– Mary Star
Nov 24 at 8:38
1
Yes, since $tau_i$ maps a root of a polynomial into a root of the same polynomial. With this claim you can show that $tau_i(a)$ define all' the roots of $min(F,a)$, since it's degree is exactly $r$, so It must have the form you have described above
– Bilo
Nov 24 at 14:57
Ah ok!! Could you explain to me also further the second point?
– Mary Star
Nov 24 at 16:41
1
Yes, for sure. First show that if $sigma,tau$ belong to the same class modulo $H$, then $sigma(a)=tau(a)$; Since $vert G:H vert = r$, each coset contains exacly $frac{n}{r}$ elements. Denote with $C_1,...,C_r$ the distinct cosets of $G$ modulo $H$., then begin{align} prod_{sigma in G}(x-tau_i(a)) &= prod_{sigma in C_1}(x-tau_1(a)) cdots prod_{sigma in C_r}(x-tau_r(a)) \ &= (x-tau_1(a))^{n/r} cdots (x-tau_r(a))^{n/r} \ &= (prod_{1}^r(x-tau_i(a)) )^{n/r} \ &= min(F,a)^{n/r} end{align} Hope it's clear.
– Bilo
Nov 24 at 17:19
1
1
Do I have to write a full proof?
– Bilo
Nov 23 at 20:39
Do I have to write a full proof?
– Bilo
Nov 23 at 20:39
For the first part: Do they have to be different because the number of the left costes is equal to the extension degree $[F(a):F]$ ?
– Mary Star
Nov 24 at 8:38
For the first part: Do they have to be different because the number of the left costes is equal to the extension degree $[F(a):F]$ ?
– Mary Star
Nov 24 at 8:38
1
1
Yes, since $tau_i$ maps a root of a polynomial into a root of the same polynomial. With this claim you can show that $tau_i(a)$ define all' the roots of $min(F,a)$, since it's degree is exactly $r$, so It must have the form you have described above
– Bilo
Nov 24 at 14:57
Yes, since $tau_i$ maps a root of a polynomial into a root of the same polynomial. With this claim you can show that $tau_i(a)$ define all' the roots of $min(F,a)$, since it's degree is exactly $r$, so It must have the form you have described above
– Bilo
Nov 24 at 14:57
Ah ok!! Could you explain to me also further the second point?
– Mary Star
Nov 24 at 16:41
Ah ok!! Could you explain to me also further the second point?
– Mary Star
Nov 24 at 16:41
1
1
Yes, for sure. First show that if $sigma,tau$ belong to the same class modulo $H$, then $sigma(a)=tau(a)$; Since $vert G:H vert = r$, each coset contains exacly $frac{n}{r}$ elements. Denote with $C_1,...,C_r$ the distinct cosets of $G$ modulo $H$., then begin{align} prod_{sigma in G}(x-tau_i(a)) &= prod_{sigma in C_1}(x-tau_1(a)) cdots prod_{sigma in C_r}(x-tau_r(a)) \ &= (x-tau_1(a))^{n/r} cdots (x-tau_r(a))^{n/r} \ &= (prod_{1}^r(x-tau_i(a)) )^{n/r} \ &= min(F,a)^{n/r} end{align} Hope it's clear.
– Bilo
Nov 24 at 17:19
Yes, for sure. First show that if $sigma,tau$ belong to the same class modulo $H$, then $sigma(a)=tau(a)$; Since $vert G:H vert = r$, each coset contains exacly $frac{n}{r}$ elements. Denote with $C_1,...,C_r$ the distinct cosets of $G$ modulo $H$., then begin{align} prod_{sigma in G}(x-tau_i(a)) &= prod_{sigma in C_1}(x-tau_1(a)) cdots prod_{sigma in C_r}(x-tau_r(a)) \ &= (x-tau_1(a))^{n/r} cdots (x-tau_r(a))^{n/r} \ &= (prod_{1}^r(x-tau_i(a)) )^{n/r} \ &= min(F,a)^{n/r} end{align} Hope it's clear.
– Bilo
Nov 24 at 17:19
|
show 1 more comment
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005655%2fgalois-extension-minimum-polynomial%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
