Mixing
Number Fields with Cyclic Galois Groups
0
$begingroup$
Let $xiinoverline{mathbb{Q}}backslashmathbb{Q}$
be an algebraic number of degree $dgeq2$
and let $mathbb{K}=mathbb{Q}left(xiright)$.
What are some conditions on $xi$
and $d$
which will guarantee that the galois group $textrm{Gal}left(mathbb{K}/mathbb{Q}right)$
of $mathbb{K}$
over $mathbb{Q}$
is cyclic? A list of multiple such conditions would be fine, assuming that there isn't already a comprehensive answer to this question.
field-theory galois-theory cyclic-groups
asked Jan 9 at 1:09
MCSMCS
969313
$endgroup$
|
show 6 more comments
0
$begingroup$
Let $xiinoverline{mathbb{Q}}backslashmathbb{Q}$
be an algebraic number of degree $dgeq2$
and let $mathbb{K}=mathbb{Q}left(xiright)$.
What are some conditions on $xi$
and $d$
which will guarantee that the galois group $textrm{Gal}left(mathbb{K}/mathbb{Q}right)$
of $mathbb{K}$
over $mathbb{Q}$
is cyclic? A list of multiple such conditions would be fine, assuming that there isn't already a comprehensive answer to this question.
field-theory galois-theory cyclic-groups
asked Jan 9 at 1:09
MCSMCS
969313
$endgroup$
$begingroup$
possible duplicate of math.stackexchange.com/questions/2204162/…
$endgroup$
– mouthetics
Jan 9 at 1:36
1
$begingroup$
What you want to use is the fact that every abelian extension of $Bbb Q$ is contained in a cyclotomic extension $Bbb Q(zeta_n)$, where $zeta_n$ is a primitive $n$-th root of unity. There are wrinkles, but the task is not hard.
$endgroup$
– Lubin
Jan 9 at 2:31
$begingroup$
I am an analyst. I have no idea how to do any of the things of which you speak. I am looking for an answer so I can press forward. I don't like being stuck like this, unable to do anything until someone finally deigns to answer my question, but I have no other option.
$endgroup$
– MCS
Jan 9 at 20:31
1
$begingroup$
As it is presented, your question cannot be answered, because the given data are too "fuzzy". For instance, you define your number field as $K=Q(x)$. But such a primitive element $x$ is not unique. To make this point clearer, just start from the opposite point of view, take a cyclic $K$ and try to find an adequate $x$. You are immediately brought back to such trials as the one suggested by @Lubin, namely use the Kronecker-Weber theorem. But even this powerful tool couldn't always give you a canonical answer. Unless you particularize the situation by adding further hypotheses...
$endgroup$
– nguyen quang do
Jan 10 at 7:59
1
$begingroup$
This approach was suggested by @Lubin. Precisely, the "wrinkles" he alludes to can be smoothed by the theory of Gauss periods (see Wiki.)
$endgroup$
– nguyen quang do
Jan 11 at 21:30
|
show 6 more comments
0
0
0
$begingroup$
Let $xiinoverline{mathbb{Q}}backslashmathbb{Q}$
be an algebraic number of degree $dgeq2$
and let $mathbb{K}=mathbb{Q}left(xiright)$.
What are some conditions on $xi$
and $d$
which will guarantee that the galois group $textrm{Gal}left(mathbb{K}/mathbb{Q}right)$
of $mathbb{K}$
over $mathbb{Q}$
is cyclic? A list of multiple such conditions would be fine, assuming that there isn't already a comprehensive answer to this question.
field-theory galois-theory cyclic-groups
asked Jan 9 at 1:09
MCSMCS
969313
$endgroup$
Let $xiinoverline{mathbb{Q}}backslashmathbb{Q}$
be an algebraic number of degree $dgeq2$
and let $mathbb{K}=mathbb{Q}left(xiright)$.
What are some conditions on $xi$
and $d$
which will guarantee that the galois group $textrm{Gal}left(mathbb{K}/mathbb{Q}right)$
of $mathbb{K}$
over $mathbb{Q}$
is cyclic? A list of multiple such conditions would be fine, assuming that there isn't already a comprehensive answer to this question.
field-theory galois-theory cyclic-groups
field-theory galois-theory cyclic-groups
asked Jan 9 at 1:09
MCSMCS
969313
asked Jan 9 at 1:09
MCSMCS
969313
asked Jan 9 at 1:09
MCSMCS
969313
asked Jan 9 at 1:09
MCSMCS
969313
asked Jan 9 at 1:09
MCSMCS
969313
969313
$begingroup$
possible duplicate of math.stackexchange.com/questions/2204162/…
$endgroup$
– mouthetics
Jan 9 at 1:36
1
$begingroup$
What you want to use is the fact that every abelian extension of $Bbb Q$ is contained in a cyclotomic extension $Bbb Q(zeta_n)$, where $zeta_n$ is a primitive $n$-th root of unity. There are wrinkles, but the task is not hard.
$endgroup$
– Lubin
Jan 9 at 2:31
$begingroup$
I am an analyst. I have no idea how to do any of the things of which you speak. I am looking for an answer so I can press forward. I don't like being stuck like this, unable to do anything until someone finally deigns to answer my question, but I have no other option.
$endgroup$
– MCS
Jan 9 at 20:31
1
$begingroup$
As it is presented, your question cannot be answered, because the given data are too "fuzzy". For instance, you define your number field as $K=Q(x)$. But such a primitive element $x$ is not unique. To make this point clearer, just start from the opposite point of view, take a cyclic $K$ and try to find an adequate $x$. You are immediately brought back to such trials as the one suggested by @Lubin, namely use the Kronecker-Weber theorem. But even this powerful tool couldn't always give you a canonical answer. Unless you particularize the situation by adding further hypotheses...
$endgroup$
– nguyen quang do
Jan 10 at 7:59
1
$begingroup$
This approach was suggested by @Lubin. Precisely, the "wrinkles" he alludes to can be smoothed by the theory of Gauss periods (see Wiki.)
$endgroup$
– nguyen quang do
Jan 11 at 21:30
|
show 6 more comments
$begingroup$
possible duplicate of math.stackexchange.com/questions/2204162/…
$endgroup$
– mouthetics
Jan 9 at 1:36
1
$begingroup$
What you want to use is the fact that every abelian extension of $Bbb Q$ is contained in a cyclotomic extension $Bbb Q(zeta_n)$, where $zeta_n$ is a primitive $n$-th root of unity. There are wrinkles, but the task is not hard.
$endgroup$
– Lubin
Jan 9 at 2:31
$begingroup$
I am an analyst. I have no idea how to do any of the things of which you speak. I am looking for an answer so I can press forward. I don't like being stuck like this, unable to do anything until someone finally deigns to answer my question, but I have no other option.
$endgroup$
– MCS
Jan 9 at 20:31
1
$begingroup$
As it is presented, your question cannot be answered, because the given data are too "fuzzy". For instance, you define your number field as $K=Q(x)$. But such a primitive element $x$ is not unique. To make this point clearer, just start from the opposite point of view, take a cyclic $K$ and try to find an adequate $x$. You are immediately brought back to such trials as the one suggested by @Lubin, namely use the Kronecker-Weber theorem. But even this powerful tool couldn't always give you a canonical answer. Unless you particularize the situation by adding further hypotheses...
$endgroup$
– nguyen quang do
Jan 10 at 7:59
1
$begingroup$
This approach was suggested by @Lubin. Precisely, the "wrinkles" he alludes to can be smoothed by the theory of Gauss periods (see Wiki.)
$endgroup$
– nguyen quang do
Jan 11 at 21:30
$begingroup$
possible duplicate of math.stackexchange.com/questions/2204162/…
$endgroup$
– mouthetics
Jan 9 at 1:36
$begingroup$
possible duplicate of math.stackexchange.com/questions/2204162/…
$endgroup$
– mouthetics
Jan 9 at 1:36
1
1
$begingroup$
What you want to use is the fact that every abelian extension of $Bbb Q$ is contained in a cyclotomic extension $Bbb Q(zeta_n)$, where $zeta_n$ is a primitive $n$-th root of unity. There are wrinkles, but the task is not hard.
$endgroup$
– Lubin
Jan 9 at 2:31
$begingroup$
What you want to use is the fact that every abelian extension of $Bbb Q$ is contained in a cyclotomic extension $Bbb Q(zeta_n)$, where $zeta_n$ is a primitive $n$-th root of unity. There are wrinkles, but the task is not hard.
$endgroup$
– Lubin
Jan 9 at 2:31
$begingroup$
I am an analyst. I have no idea how to do any of the things of which you speak. I am looking for an answer so I can press forward. I don't like being stuck like this, unable to do anything until someone finally deigns to answer my question, but I have no other option.
$endgroup$
– MCS
Jan 9 at 20:31
$begingroup$
I am an analyst. I have no idea how to do any of the things of which you speak. I am looking for an answer so I can press forward. I don't like being stuck like this, unable to do anything until someone finally deigns to answer my question, but I have no other option.
$endgroup$
– MCS
Jan 9 at 20:31
1
1
$begingroup$
As it is presented, your question cannot be answered, because the given data are too "fuzzy". For instance, you define your number field as $K=Q(x)$. But such a primitive element $x$ is not unique. To make this point clearer, just start from the opposite point of view, take a cyclic $K$ and try to find an adequate $x$. You are immediately brought back to such trials as the one suggested by @Lubin, namely use the Kronecker-Weber theorem. But even this powerful tool couldn't always give you a canonical answer. Unless you particularize the situation by adding further hypotheses...
$endgroup$
– nguyen quang do
Jan 10 at 7:59
$begingroup$
As it is presented, your question cannot be answered, because the given data are too "fuzzy". For instance, you define your number field as $K=Q(x)$. But such a primitive element $x$ is not unique. To make this point clearer, just start from the opposite point of view, take a cyclic $K$ and try to find an adequate $x$. You are immediately brought back to such trials as the one suggested by @Lubin, namely use the Kronecker-Weber theorem. But even this powerful tool couldn't always give you a canonical answer. Unless you particularize the situation by adding further hypotheses...
$endgroup$
– nguyen quang do
Jan 10 at 7:59
1
1
$begingroup$
This approach was suggested by @Lubin. Precisely, the "wrinkles" he alludes to can be smoothed by the theory of Gauss periods (see Wiki.)
$endgroup$
– nguyen quang do
Jan 11 at 21:30
$begingroup$
This approach was suggested by @Lubin. Precisely, the "wrinkles" he alludes to can be smoothed by the theory of Gauss periods (see Wiki.)
$endgroup$
– nguyen quang do
Jan 11 at 21:30
|
show 6 more comments
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$begingroup$
possible duplicate of math.stackexchange.com/questions/2204162/…
$endgroup$
– mouthetics
Jan 9 at 1:36
1
$begingroup$
What you want to use is the fact that every abelian extension of $Bbb Q$ is contained in a cyclotomic extension $Bbb Q(zeta_n)$, where $zeta_n$ is a primitive $n$-th root of unity. There are wrinkles, but the task is not hard.
$endgroup$
– Lubin
Jan 9 at 2:31
$begingroup$
I am an analyst. I have no idea how to do any of the things of which you speak. I am looking for an answer so I can press forward. I don't like being stuck like this, unable to do anything until someone finally deigns to answer my question, but I have no other option.
$endgroup$
– MCS
Jan 9 at 20:31
1
$begingroup$
As it is presented, your question cannot be answered, because the given data are too "fuzzy". For instance, you define your number field as $K=Q(x)$. But such a primitive element $x$ is not unique. To make this point clearer, just start from the opposite point of view, take a cyclic $K$ and try to find an adequate $x$. You are immediately brought back to such trials as the one suggested by @Lubin, namely use the Kronecker-Weber theorem. But even this powerful tool couldn't always give you a canonical answer. Unless you particularize the situation by adding further hypotheses...
$endgroup$
– nguyen quang do
Jan 10 at 7:59
1
$begingroup$
This approach was suggested by @Lubin. Precisely, the "wrinkles" he alludes to can be smoothed by the theory of Gauss periods (see Wiki.)
$endgroup$
– nguyen quang do
Jan 11 at 21:30