Proof: $ { mathbb{Q}[x] } / (x^2 +3x +1) cong mathbb{Q}(sqrt{5}) $
I have a problem with proving this:
$$
mathbb{Q}[x] / (x^2 +3x +1) cong mathbb{Q}(sqrt{5})
$$
My observations:
In $$mathbb{Q}[x] / (x^2 +3x +1)$$ $$x^2 equiv -1 -3x$$ so $$forall f in mathbb{Q}[x]/(x^2 +3x +1) qquad(deg(f) le 1).$$
Also, we can see that
$$x^2 + 3x +1 = (x+frac 32 +sqrt5)(x+frac 32-sqrt5)$$
And then I got stuck.
abstract-algebra ring-theory field-theory ring-isomorphism
edited Nov 24 at 23:38
Clayton
18.9k33085
asked Nov 24 at 23:12
Sharlotta Neimor
846
add a comment |
I have a problem with proving this:
$$
mathbb{Q}[x] / (x^2 +3x +1) cong mathbb{Q}(sqrt{5})
$$
My observations:
In $$mathbb{Q}[x] / (x^2 +3x +1)$$ $$x^2 equiv -1 -3x$$ so $$forall f in mathbb{Q}[x]/(x^2 +3x +1) qquad(deg(f) le 1).$$
Also, we can see that
$$x^2 + 3x +1 = (x+frac 32 +sqrt5)(x+frac 32-sqrt5)$$
And then I got stuck.
abstract-algebra ring-theory field-theory ring-isomorphism
edited Nov 24 at 23:38
Clayton
18.9k33085
asked Nov 24 at 23:12
Sharlotta Neimor
846
add a comment |
I have a problem with proving this:
$$
mathbb{Q}[x] / (x^2 +3x +1) cong mathbb{Q}(sqrt{5})
$$
My observations:
In $$mathbb{Q}[x] / (x^2 +3x +1)$$ $$x^2 equiv -1 -3x$$ so $$forall f in mathbb{Q}[x]/(x^2 +3x +1) qquad(deg(f) le 1).$$
Also, we can see that
$$x^2 + 3x +1 = (x+frac 32 +sqrt5)(x+frac 32-sqrt5)$$
And then I got stuck.
abstract-algebra ring-theory field-theory ring-isomorphism
edited Nov 24 at 23:38
Clayton
18.9k33085
asked Nov 24 at 23:12
Sharlotta Neimor
846
I have a problem with proving this:
$$
mathbb{Q}[x] / (x^2 +3x +1) cong mathbb{Q}(sqrt{5})
$$
My observations:
In $$mathbb{Q}[x] / (x^2 +3x +1)$$ $$x^2 equiv -1 -3x$$ so $$forall f in mathbb{Q}[x]/(x^2 +3x +1) qquad(deg(f) le 1).$$
Also, we can see that
$$x^2 + 3x +1 = (x+frac 32 +sqrt5)(x+frac 32-sqrt5)$$
And then I got stuck.
abstract-algebra ring-theory field-theory ring-isomorphism
abstract-algebra ring-theory field-theory ring-isomorphism
edited Nov 24 at 23:38
Clayton
18.9k33085
asked Nov 24 at 23:12
Sharlotta Neimor
846
edited Nov 24 at 23:38
Clayton
18.9k33085
asked Nov 24 at 23:12
Sharlotta Neimor
846
edited Nov 24 at 23:38
Clayton
18.9k33085
edited Nov 24 at 23:38
Clayton
18.9k33085
edited Nov 24 at 23:38
Clayton
18.9k33085
18.9k33085
asked Nov 24 at 23:12
Sharlotta Neimor
846
asked Nov 24 at 23:12
Sharlotta Neimor
846
asked Nov 24 at 23:12
Sharlotta Neimor
846
846
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
A root of $x^2+3x+1$ is $(sqrt{5}-3)/2$, so
$$
mathbb{Q}[x]big/(x^2+3x+1)congmathbb{Q}bigl((sqrt{5}-3)/2bigr)
$$
Observe that
$$
sqrt{5}=3+2frac{sqrt{5}-3}{2}
$$
and you're done.
In general, if $fin F[x]$ is an irreducible polynomial over the field $F$ and $E$ is an extension field of $F$ such that $f(alpha)=0$, for some $alphain K$, then
$$
F[x]/(f)cong F(alpha)
$$
Consider the homomorphism $varphicolon F[x]to K$ which is the identity on $F$ and sends $x$ to $alpha$. Then the image of $varphi$ is $F[alpha]$, the smallest subring of $K$ containing $F$ and $alpha$.
Show that $kervarphi=(f)$ and this will give
$$
F[x]/(f)cong F[alpha]
$$
On the other hand, $(f)$ is a maximal ideal in $F[x]$, so $F[alpha]$ is a field and therefore $F[alpha]=F(alpha)$, the smallest subfield of $K$ containing $F$ and $alpha$.
answered Nov 24 at 23:18
egreg
178k1484201
can you explain please, why ${mathbb{Q}[x]/(x^2 + 3x +1) cong mathbb(Q)((sqrt5 -3)/2)$ ?
– Sharlotta Neimor
Nov 24 at 23:24
1
@SharlottaNeimor It's a standard result on field extensions: if $f$ is an irreducible polynomial over a field $F$, then $F[x]/(f)cong F(alpha)$, where $alpha$ is any root of $f$ in some extension field of $F$.
– egreg
Nov 24 at 23:28
can you recommend me any resources where I can find the proof or may I ask you the theorem title?
– Sharlotta Neimor
Nov 24 at 23:30
1
@SharlottaNeimor I added a sketch of the proof.
– egreg
Nov 24 at 23:36
thank you so match
– Sharlotta Neimor
Nov 24 at 23:59
add a 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',
autoActivateHeartbeat: false,
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%2f3012220%2fproof-mathbbqx-x2-3x-1-cong-mathbbq-sqrt5%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
A root of $x^2+3x+1$ is $(sqrt{5}-3)/2$, so
$$
mathbb{Q}[x]big/(x^2+3x+1)congmathbb{Q}bigl((sqrt{5}-3)/2bigr)
$$
Observe that
$$
sqrt{5}=3+2frac{sqrt{5}-3}{2}
$$
and you're done.
In general, if $fin F[x]$ is an irreducible polynomial over the field $F$ and $E$ is an extension field of $F$ such that $f(alpha)=0$, for some $alphain K$, then
$$
F[x]/(f)cong F(alpha)
$$
Consider the homomorphism $varphicolon F[x]to K$ which is the identity on $F$ and sends $x$ to $alpha$. Then the image of $varphi$ is $F[alpha]$, the smallest subring of $K$ containing $F$ and $alpha$.
Show that $kervarphi=(f)$ and this will give
$$
F[x]/(f)cong F[alpha]
$$
On the other hand, $(f)$ is a maximal ideal in $F[x]$, so $F[alpha]$ is a field and therefore $F[alpha]=F(alpha)$, the smallest subfield of $K$ containing $F$ and $alpha$.
answered Nov 24 at 23:18
egreg
178k1484201
can you explain please, why ${mathbb{Q}[x]/(x^2 + 3x +1) cong mathbb(Q)((sqrt5 -3)/2)$ ?
– Sharlotta Neimor
Nov 24 at 23:24
1
@SharlottaNeimor It's a standard result on field extensions: if $f$ is an irreducible polynomial over a field $F$, then $F[x]/(f)cong F(alpha)$, where $alpha$ is any root of $f$ in some extension field of $F$.
– egreg
Nov 24 at 23:28
can you recommend me any resources where I can find the proof or may I ask you the theorem title?
– Sharlotta Neimor
Nov 24 at 23:30
1
@SharlottaNeimor I added a sketch of the proof.
– egreg
Nov 24 at 23:36
thank you so match
– Sharlotta Neimor
Nov 24 at 23:59
add a comment |
A root of $x^2+3x+1$ is $(sqrt{5}-3)/2$, so
$$
mathbb{Q}[x]big/(x^2+3x+1)congmathbb{Q}bigl((sqrt{5}-3)/2bigr)
$$
Observe that
$$
sqrt{5}=3+2frac{sqrt{5}-3}{2}
$$
and you're done.
In general, if $fin F[x]$ is an irreducible polynomial over the field $F$ and $E$ is an extension field of $F$ such that $f(alpha)=0$, for some $alphain K$, then
$$
F[x]/(f)cong F(alpha)
$$
Consider the homomorphism $varphicolon F[x]to K$ which is the identity on $F$ and sends $x$ to $alpha$. Then the image of $varphi$ is $F[alpha]$, the smallest subring of $K$ containing $F$ and $alpha$.
Show that $kervarphi=(f)$ and this will give
$$
F[x]/(f)cong F[alpha]
$$
On the other hand, $(f)$ is a maximal ideal in $F[x]$, so $F[alpha]$ is a field and therefore $F[alpha]=F(alpha)$, the smallest subfield of $K$ containing $F$ and $alpha$.
answered Nov 24 at 23:18
egreg
178k1484201
can you explain please, why ${mathbb{Q}[x]/(x^2 + 3x +1) cong mathbb(Q)((sqrt5 -3)/2)$ ?
– Sharlotta Neimor
Nov 24 at 23:24
1
@SharlottaNeimor It's a standard result on field extensions: if $f$ is an irreducible polynomial over a field $F$, then $F[x]/(f)cong F(alpha)$, where $alpha$ is any root of $f$ in some extension field of $F$.
– egreg
Nov 24 at 23:28
can you recommend me any resources where I can find the proof or may I ask you the theorem title?
– Sharlotta Neimor
Nov 24 at 23:30
1
@SharlottaNeimor I added a sketch of the proof.
– egreg
Nov 24 at 23:36
thank you so match
– Sharlotta Neimor
Nov 24 at 23:59
add a comment |
A root of $x^2+3x+1$ is $(sqrt{5}-3)/2$, so
$$
mathbb{Q}[x]big/(x^2+3x+1)congmathbb{Q}bigl((sqrt{5}-3)/2bigr)
$$
Observe that
$$
sqrt{5}=3+2frac{sqrt{5}-3}{2}
$$
and you're done.
In general, if $fin F[x]$ is an irreducible polynomial over the field $F$ and $E$ is an extension field of $F$ such that $f(alpha)=0$, for some $alphain K$, then
$$
F[x]/(f)cong F(alpha)
$$
Consider the homomorphism $varphicolon F[x]to K$ which is the identity on $F$ and sends $x$ to $alpha$. Then the image of $varphi$ is $F[alpha]$, the smallest subring of $K$ containing $F$ and $alpha$.
Show that $kervarphi=(f)$ and this will give
$$
F[x]/(f)cong F[alpha]
$$
On the other hand, $(f)$ is a maximal ideal in $F[x]$, so $F[alpha]$ is a field and therefore $F[alpha]=F(alpha)$, the smallest subfield of $K$ containing $F$ and $alpha$.
answered Nov 24 at 23:18
egreg
178k1484201
A root of $x^2+3x+1$ is $(sqrt{5}-3)/2$, so
$$
mathbb{Q}[x]big/(x^2+3x+1)congmathbb{Q}bigl((sqrt{5}-3)/2bigr)
$$
Observe that
$$
sqrt{5}=3+2frac{sqrt{5}-3}{2}
$$
and you're done.
In general, if $fin F[x]$ is an irreducible polynomial over the field $F$ and $E$ is an extension field of $F$ such that $f(alpha)=0$, for some $alphain K$, then
$$
F[x]/(f)cong F(alpha)
$$
Consider the homomorphism $varphicolon F[x]to K$ which is the identity on $F$ and sends $x$ to $alpha$. Then the image of $varphi$ is $F[alpha]$, the smallest subring of $K$ containing $F$ and $alpha$.
Show that $kervarphi=(f)$ and this will give
$$
F[x]/(f)cong F[alpha]
$$
On the other hand, $(f)$ is a maximal ideal in $F[x]$, so $F[alpha]$ is a field and therefore $F[alpha]=F(alpha)$, the smallest subfield of $K$ containing $F$ and $alpha$.
answered Nov 24 at 23:18
egreg
178k1484201
edited Nov 24 at 23:36
answered Nov 24 at 23:18
egreg
178k1484201
answered Nov 24 at 23:18
egreg
178k1484201
answered Nov 24 at 23:18
egreg
178k1484201
178k1484201
can you explain please, why ${mathbb{Q}[x]/(x^2 + 3x +1) cong mathbb(Q)((sqrt5 -3)/2)$ ?
– Sharlotta Neimor
Nov 24 at 23:24
1
@SharlottaNeimor It's a standard result on field extensions: if $f$ is an irreducible polynomial over a field $F$, then $F[x]/(f)cong F(alpha)$, where $alpha$ is any root of $f$ in some extension field of $F$.
– egreg
Nov 24 at 23:28
can you recommend me any resources where I can find the proof or may I ask you the theorem title?
– Sharlotta Neimor
Nov 24 at 23:30
1
@SharlottaNeimor I added a sketch of the proof.
– egreg
Nov 24 at 23:36
thank you so match
– Sharlotta Neimor
Nov 24 at 23:59
add a comment |
can you explain please, why ${mathbb{Q}[x]/(x^2 + 3x +1) cong mathbb(Q)((sqrt5 -3)/2)$ ?
– Sharlotta Neimor
Nov 24 at 23:24
1
@SharlottaNeimor It's a standard result on field extensions: if $f$ is an irreducible polynomial over a field $F$, then $F[x]/(f)cong F(alpha)$, where $alpha$ is any root of $f$ in some extension field of $F$.
– egreg
Nov 24 at 23:28
can you recommend me any resources where I can find the proof or may I ask you the theorem title?
– Sharlotta Neimor
Nov 24 at 23:30
1
@SharlottaNeimor I added a sketch of the proof.
– egreg
Nov 24 at 23:36
thank you so match
– Sharlotta Neimor
Nov 24 at 23:59
can you explain please, why ${mathbb{Q}[x]/(x^2 + 3x +1) cong mathbb(Q)((sqrt5 -3)/2)$ ?
– Sharlotta Neimor
Nov 24 at 23:24
can you explain please, why ${mathbb{Q}[x]/(x^2 + 3x +1) cong mathbb(Q)((sqrt5 -3)/2)$ ?
– Sharlotta Neimor
Nov 24 at 23:24
1
1
@SharlottaNeimor It's a standard result on field extensions: if $f$ is an irreducible polynomial over a field $F$, then $F[x]/(f)cong F(alpha)$, where $alpha$ is any root of $f$ in some extension field of $F$.
– egreg
Nov 24 at 23:28
@SharlottaNeimor It's a standard result on field extensions: if $f$ is an irreducible polynomial over a field $F$, then $F[x]/(f)cong F(alpha)$, where $alpha$ is any root of $f$ in some extension field of $F$.
– egreg
Nov 24 at 23:28
can you recommend me any resources where I can find the proof or may I ask you the theorem title?
– Sharlotta Neimor
Nov 24 at 23:30
can you recommend me any resources where I can find the proof or may I ask you the theorem title?
– Sharlotta Neimor
Nov 24 at 23:30
1
1
@SharlottaNeimor I added a sketch of the proof.
– egreg
Nov 24 at 23:36
@SharlottaNeimor I added a sketch of the proof.
– egreg
Nov 24 at 23:36
thank you so match
– Sharlotta Neimor
Nov 24 at 23:59
thank you so match
– Sharlotta Neimor
Nov 24 at 23:59
add a 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%2f3012220%2fproof-mathbbqx-x2-3x-1-cong-mathbbq-sqrt5%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
