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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 $mathbb{F}_5$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













4












$begingroup$


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 $mathbb{Q}$ 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?










share|cite|improve this question









$endgroup$








  • 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
    46 mins ago


















4












$begingroup$


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 $mathbb{Q}$ 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?










share|cite|improve this question









$endgroup$








  • 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
    46 mins ago
















4












4








4


0



$begingroup$


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 $mathbb{Q}$ 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?










share|cite|improve this question









$endgroup$




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 $mathbb{Q}$ 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






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asked 1 hour 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
    46 mins ago
















  • 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
    46 mins 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
46 mins 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
46 mins ago












1 Answer
1






active

oldest

votes


















5












$begingroup$

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 $F(theta)$.






share|cite|improve this answer









$endgroup$









  • 2




    $begingroup$
    And those powers of $theta$ are distinct elements of the field.
    $endgroup$
    – Gerry Myerson
    39 mins ago











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









5












$begingroup$

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 $F(theta)$.






share|cite|improve this answer









$endgroup$









  • 2




    $begingroup$
    And those powers of $theta$ are distinct elements of the field.
    $endgroup$
    – Gerry Myerson
    39 mins ago
















5












$begingroup$

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 $F(theta)$.






share|cite|improve this answer









$endgroup$









  • 2




    $begingroup$
    And those powers of $theta$ are distinct elements of the field.
    $endgroup$
    – Gerry Myerson
    39 mins ago














5












5








5





$begingroup$

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 $F(theta)$.






share|cite|improve this answer









$endgroup$



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 $F(theta)$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 52 mins ago









lhflhf

166k10171400




166k10171400








  • 2




    $begingroup$
    And those powers of $theta$ are distinct elements of the field.
    $endgroup$
    – Gerry Myerson
    39 mins ago














  • 2




    $begingroup$
    And those powers of $theta$ are distinct elements of the field.
    $endgroup$
    – Gerry Myerson
    39 mins ago








2




2




$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
39 mins ago




$begingroup$
And those powers of $theta$ are distinct elements of the field.
$endgroup$
– Gerry Myerson
39 mins ago


















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