Understanding of a proof
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In (1) and (3), $ M' subseteq L'$ and $H subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' leq L'$ and $H leq H''$ again respectively.
galois-theory extension-field
asked Dec 30 '18 at 0:20
Leyla AlkanLeyla Alkan
1,5401724
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add a comment |
$begingroup$
In (1) and (3), $ M' subseteq L'$ and $H subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' leq L'$ and $H leq H''$ again respectively.
galois-theory extension-field
asked Dec 30 '18 at 0:20
Leyla AlkanLeyla Alkan
1,5401724
$endgroup$
add a comment |
$begingroup$
In (1) and (3), $ M' subseteq L'$ and $H subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' leq L'$ and $H leq H''$ again respectively.
galois-theory extension-field
asked Dec 30 '18 at 0:20
Leyla AlkanLeyla Alkan
1,5401724
$endgroup$
In (1) and (3), $ M' subseteq L'$ and $H subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' leq L'$ and $H leq H''$ again respectively.
galois-theory extension-field
galois-theory extension-field
asked Dec 30 '18 at 0:20
Leyla AlkanLeyla Alkan
1,5401724
asked Dec 30 '18 at 0:20
Leyla AlkanLeyla Alkan
1,5401724
asked Dec 30 '18 at 0:20
Leyla AlkanLeyla Alkan
1,5401724
asked Dec 30 '18 at 0:20
Leyla AlkanLeyla Alkan
1,5401724
asked Dec 30 '18 at 0:20
Leyla AlkanLeyla Alkan
1,5401724
1,5401724
add a comment |
add a comment |
1 Answer
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$begingroup$
I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.
Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.
answered Dec 30 '18 at 0:35
Noble MushtakNoble Mushtak
15.4k1835
$endgroup$
$begingroup$
I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
$endgroup$
– Leyla Alkan
Dec 30 '18 at 0:45
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.
Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.
answered Dec 30 '18 at 0:35
Noble MushtakNoble Mushtak
15.4k1835
$endgroup$
$begingroup$
I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
$endgroup$
– Leyla Alkan
Dec 30 '18 at 0:45
add a comment |
$begingroup$
I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.
Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.
answered Dec 30 '18 at 0:35
Noble MushtakNoble Mushtak
15.4k1835
$endgroup$
$begingroup$
I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
$endgroup$
– Leyla Alkan
Dec 30 '18 at 0:45
add a comment |
$begingroup$
I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.
Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.
answered Dec 30 '18 at 0:35
Noble MushtakNoble Mushtak
15.4k1835
$endgroup$
I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.
Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.
answered Dec 30 '18 at 0:35
Noble MushtakNoble Mushtak
15.4k1835
answered Dec 30 '18 at 0:35
Noble MushtakNoble Mushtak
15.4k1835
answered Dec 30 '18 at 0:35
Noble MushtakNoble Mushtak
15.4k1835
answered Dec 30 '18 at 0:35
Noble MushtakNoble Mushtak
15.4k1835
15.4k1835
$begingroup$
I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
$endgroup$
– Leyla Alkan
Dec 30 '18 at 0:45
add a comment |
$begingroup$
I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
$endgroup$
– Leyla Alkan
Dec 30 '18 at 0:45
$begingroup$
I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
$endgroup$
– Leyla Alkan
Dec 30 '18 at 0:45
$begingroup$
I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
$endgroup$
– Leyla Alkan
Dec 30 '18 at 0:45
add a comment |
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