Extension of Splitting Fields over An Arbitrary FieldSplitting field implies Galois extensionWhat does it mean to take the splitting field of $f(x)in F[x]$ over $K$ where $K/F$ is a field extensionCalculating Splitting Field Degree of ExtensionDetermining whether or not an extension is a splitting fieldElementary Field Theory: Extension Field of Degree 2Splitting field of $x^3 - 2$ over $mathbbF_5$Normal field extension implies splitting fieldSplitting fields and their degreesWhat things we have to take care of while finding the degree of field extension, splitting fields for some polynomial?A question on the definition of splitting field
If the Captain's screens are out, does he switch seats with the co-pilot?
What is the dot in “1.2.4."
My adviser wants to be the first author
Need some help with my first LaTeX drawing…
Why does Deadpool say "You're welcome, Canada," after shooting Ryan Reynolds in the end credits?
What exactly is the purpose of connection links straped between the rocket and the launch pad
Playing ONE triplet (not three)
Time dilation for a moving electronic clock
How can I discourage/prevent PCs from using door choke-points?
When were linguistics departments first established
Coworker uses her breast-pump everywhere in the office
Rejected in 4th interview round citing insufficient years of experience
Plywood subfloor won't screw down in a trailer home
Time travel short story where dinosaur doesn't taste like chicken
Should QA ask requirements to developers?
Sword in the Stone story where the sword was held in place by electromagnets
Can "semicircle" be used to refer to a part-circle that is not a exact half-circle?
Touchscreen-controlled dentist office snowman collector game
"However" used in a conditional clause?
Who is our nearest neighbor
It's a yearly task, alright
Is all copper pipe pretty much the same?
How to deal with a cynical class?
Question about partial fractions with irreducible quadratic factors
Extension of Splitting Fields over An Arbitrary Field
Splitting field implies Galois extensionWhat does it mean to take the splitting field of $f(x)in F[x]$ over $K$ where $K/F$ is a field extensionCalculating Splitting Field Degree of ExtensionDetermining whether or not an extension is a splitting fieldElementary Field Theory: Extension Field of Degree 2Splitting field of $x^3 - 2$ over $mathbbF_5$Normal field extension implies splitting fieldSplitting fields and their degreesWhat things we have to take care of while finding the degree of field extension, splitting fields for some polynomial?A question on the definition of splitting field
$begingroup$
Let $F$ be a field in which $0 neq2$ in $F$, and consider $f=x^4+1$. If $E$ is the splitting field for $f$ over $F$, it turns out that $E$ is a simple extension of $F$. How does one realize this fact? I'm not so sure as to what field element I can adjoin to $F$ to allow $f$ to split into linear factors. Finding the splitting field over something like $mathbbQ$ is straight forward and easy in comparison, but I'm having trouble working with any general field $F$.
Also, if we indeed did have that $0=2$ in our field $F$, then $f=x^4+1=(x+1)^4$, so $F$ is its own splitting field, is this correct reasoning?
abstract-algebra field-theory extension-field splitting-field
asked 2 hours ago
DevilofHell'sKitchenDevilofHell'sKitchen
405
$endgroup$
add a comment |
$begingroup$
Let $F$ be a field in which $0 neq2$ in $F$, and consider $f=x^4+1$. If $E$ is the splitting field for $f$ over $F$, it turns out that $E$ is a simple extension of $F$. How does one realize this fact? I'm not so sure as to what field element I can adjoin to $F$ to allow $f$ to split into linear factors. Finding the splitting field over something like $mathbbQ$ is straight forward and easy in comparison, but I'm having trouble working with any general field $F$.
Also, if we indeed did have that $0=2$ in our field $F$, then $f=x^4+1=(x+1)^4$, so $F$ is its own splitting field, is this correct reasoning?
abstract-algebra field-theory extension-field splitting-field
asked 2 hours ago
DevilofHell'sKitchenDevilofHell'sKitchen
405
$endgroup$
2
$begingroup$
Letting $alpha$ be any root, then $f$ splits as $(x-alpha)(x+alpha)(x-alpha^3)(x+alpha^3)$ in $F[alpha]$.
$endgroup$
– Mike Earnest
2 hours ago
add a comment |
$begingroup$
Let $F$ be a field in which $0 neq2$ in $F$, and consider $f=x^4+1$. If $E$ is the splitting field for $f$ over $F$, it turns out that $E$ is a simple extension of $F$. How does one realize this fact? I'm not so sure as to what field element I can adjoin to $F$ to allow $f$ to split into linear factors. Finding the splitting field over something like $mathbbQ$ is straight forward and easy in comparison, but I'm having trouble working with any general field $F$.
Also, if we indeed did have that $0=2$ in our field $F$, then $f=x^4+1=(x+1)^4$, so $F$ is its own splitting field, is this correct reasoning?
abstract-algebra field-theory extension-field splitting-field
asked 2 hours ago
DevilofHell'sKitchenDevilofHell'sKitchen
405
$endgroup$
Let $F$ be a field in which $0 neq2$ in $F$, and consider $f=x^4+1$. If $E$ is the splitting field for $f$ over $F$, it turns out that $E$ is a simple extension of $F$. How does one realize this fact? I'm not so sure as to what field element I can adjoin to $F$ to allow $f$ to split into linear factors. Finding the splitting field over something like $mathbbQ$ is straight forward and easy in comparison, but I'm having trouble working with any general field $F$.
Also, if we indeed did have that $0=2$ in our field $F$, then $f=x^4+1=(x+1)^4$, so $F$ is its own splitting field, is this correct reasoning?
abstract-algebra field-theory extension-field splitting-field
abstract-algebra field-theory extension-field splitting-field
asked 2 hours ago
DevilofHell'sKitchenDevilofHell'sKitchen
405
asked 2 hours ago
DevilofHell'sKitchenDevilofHell'sKitchen
405
asked 2 hours ago
DevilofHell'sKitchenDevilofHell'sKitchen
405
asked 2 hours ago
DevilofHell'sKitchenDevilofHell'sKitchen
405
asked 2 hours ago
DevilofHell'sKitchenDevilofHell'sKitchen
405
405
2
$begingroup$
Letting $alpha$ be any root, then $f$ splits as $(x-alpha)(x+alpha)(x-alpha^3)(x+alpha^3)$ in $F[alpha]$.
$endgroup$
– Mike Earnest
2 hours ago
add a comment |
2
$begingroup$
Letting $alpha$ be any root, then $f$ splits as $(x-alpha)(x+alpha)(x-alpha^3)(x+alpha^3)$ in $F[alpha]$.
$endgroup$
– Mike Earnest
2 hours ago
2
2
$begingroup$
Letting $alpha$ be any root, then $f$ splits as $(x-alpha)(x+alpha)(x-alpha^3)(x+alpha^3)$ in $F[alpha]$.
$endgroup$
– Mike Earnest
2 hours ago
$begingroup$
Letting $alpha$ be any root, then $f$ splits as $(x-alpha)(x+alpha)(x-alpha^3)(x+alpha^3)$ in $F[alpha]$.
$endgroup$
– Mike Earnest
2 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If $theta$ is a root of $x^4+1$, then so are $theta^k$ for $k=1,3,5,7$, and so $x^4+1$ splits in $F(theta)$.
answered 2 hours ago
lhflhf
166k10171400
$endgroup$
2
$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
2 hours ago
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%2f3147449%2fextension-of-splitting-fields-over-an-arbitrary-field%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
$begingroup$
If $theta$ is a root of $x^4+1$, then so are $theta^k$ for $k=1,3,5,7$, and so $x^4+1$ splits in $F(theta)$.
answered 2 hours ago
lhflhf
166k10171400
$endgroup$
2
$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
2 hours ago
add a comment |
$begingroup$
If $theta$ is a root of $x^4+1$, then so are $theta^k$ for $k=1,3,5,7$, and so $x^4+1$ splits in $F(theta)$.
answered 2 hours ago
lhflhf
166k10171400
$endgroup$
2
$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
2 hours ago
add a comment |
$begingroup$
If $theta$ is a root of $x^4+1$, then so are $theta^k$ for $k=1,3,5,7$, and so $x^4+1$ splits in $F(theta)$.
answered 2 hours ago
lhflhf
166k10171400
$endgroup$
If $theta$ is a root of $x^4+1$, then so are $theta^k$ for $k=1,3,5,7$, and so $x^4+1$ splits in $F(theta)$.
answered 2 hours ago
lhflhf
166k10171400
answered 2 hours ago
lhflhf
166k10171400
answered 2 hours ago
lhflhf
166k10171400
answered 2 hours ago
lhflhf
166k10171400
166k10171400
2
$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
2 hours ago
add a comment |
2
$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
2 hours ago
2
2
$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
2 hours ago
$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
2 hours ago
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.
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%2f3147449%2fextension-of-splitting-fields-over-an-arbitrary-field%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
2
$begingroup$
Letting $alpha$ be any root, then $f$ splits as $(x-alpha)(x+alpha)(x-alpha^3)(x+alpha^3)$ in $F[alpha]$.
$endgroup$
– Mike Earnest
2 hours ago