Find the degree of the extensions $mathbb{F}(x) / mathbb{F}$
$begingroup$
Here $mathbb{F}$ is a field and $mathbb{F}(x) ={ frac{q}{p}: q,p in mathbb{F}[x], p neq 0}$.
I think the proper way to solve is to use the fact that an infinite extension of a countable field should be of infinite degree. However, $mathbb{F}$ is not necessarily countable.
abstract-algebra field-theory
edited Dec 10 '18 at 13:47
egreg
181k1485202
asked Dec 10 '18 at 12:56
XenidiaXenidia
1,275729
$endgroup$
add a comment |
$begingroup$
Here $mathbb{F}$ is a field and $mathbb{F}(x) ={ frac{q}{p}: q,p in mathbb{F}[x], p neq 0}$.
I think the proper way to solve is to use the fact that an infinite extension of a countable field should be of infinite degree. However, $mathbb{F}$ is not necessarily countable.
abstract-algebra field-theory
edited Dec 10 '18 at 13:47
egreg
181k1485202
asked Dec 10 '18 at 12:56
XenidiaXenidia
1,275729
$endgroup$
$begingroup$
This does not quite look like a fact, but rather as a tautology.
$endgroup$
– Saucy O'Path
Dec 10 '18 at 13:00
$begingroup$
The set ${1,x, x^2, ldots }$ is linearly independent.
$endgroup$
– Slade
Dec 10 '18 at 13:53
add a comment |
$begingroup$
Here $mathbb{F}$ is a field and $mathbb{F}(x) ={ frac{q}{p}: q,p in mathbb{F}[x], p neq 0}$.
I think the proper way to solve is to use the fact that an infinite extension of a countable field should be of infinite degree. However, $mathbb{F}$ is not necessarily countable.
abstract-algebra field-theory
edited Dec 10 '18 at 13:47
egreg
181k1485202
asked Dec 10 '18 at 12:56
XenidiaXenidia
1,275729
$endgroup$
Here $mathbb{F}$ is a field and $mathbb{F}(x) ={ frac{q}{p}: q,p in mathbb{F}[x], p neq 0}$.
I think the proper way to solve is to use the fact that an infinite extension of a countable field should be of infinite degree. However, $mathbb{F}$ is not necessarily countable.
abstract-algebra field-theory
abstract-algebra field-theory
edited Dec 10 '18 at 13:47
egreg
181k1485202
asked Dec 10 '18 at 12:56
XenidiaXenidia
1,275729
edited Dec 10 '18 at 13:47
egreg
181k1485202
asked Dec 10 '18 at 12:56
XenidiaXenidia
1,275729
edited Dec 10 '18 at 13:47
egreg
181k1485202
edited Dec 10 '18 at 13:47
egreg
181k1485202
edited Dec 10 '18 at 13:47
egreg
181k1485202
181k1485202
asked Dec 10 '18 at 12:56
XenidiaXenidia
1,275729
asked Dec 10 '18 at 12:56
XenidiaXenidia
1,275729
asked Dec 10 '18 at 12:56
XenidiaXenidia
1,275729
1,275729
$begingroup$
This does not quite look like a fact, but rather as a tautology.
$endgroup$
– Saucy O'Path
Dec 10 '18 at 13:00
$begingroup$
The set ${1,x, x^2, ldots }$ is linearly independent.
$endgroup$
– Slade
Dec 10 '18 at 13:53
add a comment |
$begingroup$
This does not quite look like a fact, but rather as a tautology.
$endgroup$
– Saucy O'Path
Dec 10 '18 at 13:00
$begingroup$
The set ${1,x, x^2, ldots }$ is linearly independent.
$endgroup$
– Slade
Dec 10 '18 at 13:53
$begingroup$
This does not quite look like a fact, but rather as a tautology.
$endgroup$
– Saucy O'Path
Dec 10 '18 at 13:00
$begingroup$
This does not quite look like a fact, but rather as a tautology.
$endgroup$
– Saucy O'Path
Dec 10 '18 at 13:00
$begingroup$
The set ${1,x, x^2, ldots }$ is linearly independent.
$endgroup$
– Slade
Dec 10 '18 at 13:53
$begingroup$
The set ${1,x, x^2, ldots }$ is linearly independent.
$endgroup$
– Slade
Dec 10 '18 at 13:53
add a comment |
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$begingroup$
This does not quite look like a fact, but rather as a tautology.
$endgroup$
– Saucy O'Path
Dec 10 '18 at 13:00
$begingroup$
The set ${1,x, x^2, ldots }$ is linearly independent.
$endgroup$
– Slade
Dec 10 '18 at 13:53