Is every finite extension L/K separable? [duplicate]
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Example of finite field extension where root not separable
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I'm trying to determine whether this statement is true or false. However I only understand that every finite extension of a finite field is separable from another solution. (Every finite extension of a finite field is separable)
So what difference is finite or not with regards to the finite extension being separable or not?
Thank you
abstract-algebra
asked Nov 13 at 11:57
RandomPerson123
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marked as duplicate by Dietrich Burde abstract-algebra
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Nov 13 at 12:06
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This question already has an answer here:
Example of finite field extension where root not separable
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I'm trying to determine whether this statement is true or false. However I only understand that every finite extension of a finite field is separable from another solution. (Every finite extension of a finite field is separable)
So what difference is finite or not with regards to the finite extension being separable or not?
Thank you
abstract-algebra
asked Nov 13 at 11:57
RandomPerson123
81
marked as duplicate by Dietrich Burde abstract-algebra
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Nov 13 at 12:06
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
The statement is false, see for example here. See also this duplicate.
– Dietrich Burde
Nov 13 at 12:02
A field extension $L/K$ is called finite when its degree is finite, that is, when the dimension of $L$ as $K$-vector space is finite. This is always true if $L$ is a finite field, but for example $mathbb{C}/mathbb{R}$ is also a finite extension.
– quid♦
Nov 13 at 12:08
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up vote
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down vote
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This question already has an answer here:
Example of finite field extension where root not separable
1 answer
I'm trying to determine whether this statement is true or false. However I only understand that every finite extension of a finite field is separable from another solution. (Every finite extension of a finite field is separable)
So what difference is finite or not with regards to the finite extension being separable or not?
Thank you
abstract-algebra
asked Nov 13 at 11:57
RandomPerson123
81
This question already has an answer here:
Example of finite field extension where root not separable
1 answer
I'm trying to determine whether this statement is true or false. However I only understand that every finite extension of a finite field is separable from another solution. (Every finite extension of a finite field is separable)
So what difference is finite or not with regards to the finite extension being separable or not?
Thank you
This question already has an answer here:
Example of finite field extension where root not separable
1 answer
abstract-algebra
abstract-algebra
asked Nov 13 at 11:57
RandomPerson123
81
asked Nov 13 at 11:57
RandomPerson123
81
asked Nov 13 at 11:57
RandomPerson123
81
asked Nov 13 at 11:57
RandomPerson123
81
asked Nov 13 at 11:57
RandomPerson123
81
81
marked as duplicate by Dietrich Burde abstract-algebra
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Nov 13 at 12:06
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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Nov 13 at 12:06
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
The statement is false, see for example here. See also this duplicate.
– Dietrich Burde
Nov 13 at 12:02
A field extension $L/K$ is called finite when its degree is finite, that is, when the dimension of $L$ as $K$-vector space is finite. This is always true if $L$ is a finite field, but for example $mathbb{C}/mathbb{R}$ is also a finite extension.
– quid♦
Nov 13 at 12:08
add a comment |
The statement is false, see for example here. See also this duplicate.
– Dietrich Burde
Nov 13 at 12:02
A field extension $L/K$ is called finite when its degree is finite, that is, when the dimension of $L$ as $K$-vector space is finite. This is always true if $L$ is a finite field, but for example $mathbb{C}/mathbb{R}$ is also a finite extension.
– quid♦
Nov 13 at 12:08
The statement is false, see for example here. See also this duplicate.
– Dietrich Burde
Nov 13 at 12:02
The statement is false, see for example here. See also this duplicate.
– Dietrich Burde
Nov 13 at 12:02
A field extension $L/K$ is called finite when its degree is finite, that is, when the dimension of $L$ as $K$-vector space is finite. This is always true if $L$ is a finite field, but for example $mathbb{C}/mathbb{R}$ is also a finite extension.
– quid♦
Nov 13 at 12:08
A field extension $L/K$ is called finite when its degree is finite, that is, when the dimension of $L$ as $K$-vector space is finite. This is always true if $L$ is a finite field, but for example $mathbb{C}/mathbb{R}$ is also a finite extension.
– quid♦
Nov 13 at 12:08
add a comment |
1 Answer
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The difference is that a finite field is perfect, but there are infinite fields that aren't.
The basic example is the following (in a sense it's the "only" example) : assume you have a field $K$ of characteristic $p$, and $ain K$ has no $p$th root. Then $X^p - a$ is irreducible over $K$ (this is not immediate but follows with a bit of work).
However if you take a finite extension $L/K$ where it has a root $b$, then $X^p-a = (X-b)^p$, and so the extension isn't separable
answered Nov 13 at 12:03
Max
12.1k11038
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The difference is that a finite field is perfect, but there are infinite fields that aren't.
The basic example is the following (in a sense it's the "only" example) : assume you have a field $K$ of characteristic $p$, and $ain K$ has no $p$th root. Then $X^p - a$ is irreducible over $K$ (this is not immediate but follows with a bit of work).
However if you take a finite extension $L/K$ where it has a root $b$, then $X^p-a = (X-b)^p$, and so the extension isn't separable
answered Nov 13 at 12:03
Max
12.1k11038
add a comment |
up vote
1
down vote
accepted
The difference is that a finite field is perfect, but there are infinite fields that aren't.
The basic example is the following (in a sense it's the "only" example) : assume you have a field $K$ of characteristic $p$, and $ain K$ has no $p$th root. Then $X^p - a$ is irreducible over $K$ (this is not immediate but follows with a bit of work).
However if you take a finite extension $L/K$ where it has a root $b$, then $X^p-a = (X-b)^p$, and so the extension isn't separable
answered Nov 13 at 12:03
Max
12.1k11038
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The difference is that a finite field is perfect, but there are infinite fields that aren't.
The basic example is the following (in a sense it's the "only" example) : assume you have a field $K$ of characteristic $p$, and $ain K$ has no $p$th root. Then $X^p - a$ is irreducible over $K$ (this is not immediate but follows with a bit of work).
However if you take a finite extension $L/K$ where it has a root $b$, then $X^p-a = (X-b)^p$, and so the extension isn't separable
answered Nov 13 at 12:03
Max
12.1k11038
The difference is that a finite field is perfect, but there are infinite fields that aren't.
The basic example is the following (in a sense it's the "only" example) : assume you have a field $K$ of characteristic $p$, and $ain K$ has no $p$th root. Then $X^p - a$ is irreducible over $K$ (this is not immediate but follows with a bit of work).
However if you take a finite extension $L/K$ where it has a root $b$, then $X^p-a = (X-b)^p$, and so the extension isn't separable
answered Nov 13 at 12:03
Max
12.1k11038
answered Nov 13 at 12:03
Max
12.1k11038
answered Nov 13 at 12:03
Max
12.1k11038
answered Nov 13 at 12:03
Max
12.1k11038
12.1k11038
add a comment |
add a comment |
The statement is false, see for example here. See also this duplicate.
– Dietrich Burde
Nov 13 at 12:02
A field extension $L/K$ is called finite when its degree is finite, that is, when the dimension of $L$ as $K$-vector space is finite. This is always true if $L$ is a finite field, but for example $mathbb{C}/mathbb{R}$ is also a finite extension.
– quid♦
Nov 13 at 12:08