Norm of the ring
2
$begingroup$
What is the norm of
$$mathbb{Z}[zeta] = [a_0 + a_1zeta + ...+a_{l-2}zeta^{l-2} : a_i in mathbb{Z}],$$
for $(l leq 19)$ prime, and $zeta$ root of unity, so $mathbb{Z}[zeta]$ is UFD.
It is easy when $l=3$, $(a+bzeta)(a+bbar{zeta}) = a^2 - ab+ b^2$
abstract-algebra
edited Dec 17 '18 at 10:43
Namaste
1
asked Aug 27 '18 at 11:59
Dogukan.NDogukan.N
112
$endgroup$
add a comment |
2
$begingroup$
What is the norm of
$$mathbb{Z}[zeta] = [a_0 + a_1zeta + ...+a_{l-2}zeta^{l-2} : a_i in mathbb{Z}],$$
for $(l leq 19)$ prime, and $zeta$ root of unity, so $mathbb{Z}[zeta]$ is UFD.
It is easy when $l=3$, $(a+bzeta)(a+bbar{zeta}) = a^2 - ab+ b^2$
abstract-algebra
edited Dec 17 '18 at 10:43
Namaste
1
asked Aug 27 '18 at 11:59
Dogukan.NDogukan.N
112
$endgroup$
$begingroup$
Not sure it is very easy : for a fixed element, would you not need to find the minimal polynomial or the conjugates of that element? For $l=3$ the conjugates were easy to find, but for larger $l$ this may be an issue.
$endgroup$
– астон вілла олоф мэллбэрг
Aug 27 '18 at 12:05
$begingroup$
Do you mean norm of an element, perhaps? (Note that UFDs may not be a Euclidean domain and so there may not be a Euclidean function)
$endgroup$
– daruma
Aug 27 '18 at 13:42
$begingroup$
@daruma, see Field norm
$endgroup$
– lhf
Dec 17 '18 at 11:06
$begingroup$
For $l=5$, see math.stackexchange.com/questions/24840/…. See also math.stackexchange.com/questions/175118/…
$endgroup$
– lhf
Dec 17 '18 at 11:07
add a comment |
2
2
2
$begingroup$
What is the norm of
$$mathbb{Z}[zeta] = [a_0 + a_1zeta + ...+a_{l-2}zeta^{l-2} : a_i in mathbb{Z}],$$
for $(l leq 19)$ prime, and $zeta$ root of unity, so $mathbb{Z}[zeta]$ is UFD.
It is easy when $l=3$, $(a+bzeta)(a+bbar{zeta}) = a^2 - ab+ b^2$
abstract-algebra
edited Dec 17 '18 at 10:43
Namaste
1
asked Aug 27 '18 at 11:59
Dogukan.NDogukan.N
112
$endgroup$
What is the norm of
$$mathbb{Z}[zeta] = [a_0 + a_1zeta + ...+a_{l-2}zeta^{l-2} : a_i in mathbb{Z}],$$
for $(l leq 19)$ prime, and $zeta$ root of unity, so $mathbb{Z}[zeta]$ is UFD.
It is easy when $l=3$, $(a+bzeta)(a+bbar{zeta}) = a^2 - ab+ b^2$
abstract-algebra
abstract-algebra
edited Dec 17 '18 at 10:43
Namaste
1
asked Aug 27 '18 at 11:59
Dogukan.NDogukan.N
112
edited Dec 17 '18 at 10:43
Namaste
1
asked Aug 27 '18 at 11:59
Dogukan.NDogukan.N
112
edited Dec 17 '18 at 10:43
Namaste
1
edited Dec 17 '18 at 10:43
Namaste
1
edited Dec 17 '18 at 10:43
Namaste
1
1
asked Aug 27 '18 at 11:59
Dogukan.NDogukan.N
112
asked Aug 27 '18 at 11:59
Dogukan.NDogukan.N
112
asked Aug 27 '18 at 11:59
Dogukan.NDogukan.N
112
112
$begingroup$
Not sure it is very easy : for a fixed element, would you not need to find the minimal polynomial or the conjugates of that element? For $l=3$ the conjugates were easy to find, but for larger $l$ this may be an issue.
$endgroup$
– астон вілла олоф мэллбэрг
Aug 27 '18 at 12:05
$begingroup$
Do you mean norm of an element, perhaps? (Note that UFDs may not be a Euclidean domain and so there may not be a Euclidean function)
$endgroup$
– daruma
Aug 27 '18 at 13:42
$begingroup$
@daruma, see Field norm
$endgroup$
– lhf
Dec 17 '18 at 11:06
$begingroup$
For $l=5$, see math.stackexchange.com/questions/24840/…. See also math.stackexchange.com/questions/175118/…
$endgroup$
– lhf
Dec 17 '18 at 11:07
add a comment |
$begingroup$
Not sure it is very easy : for a fixed element, would you not need to find the minimal polynomial or the conjugates of that element? For $l=3$ the conjugates were easy to find, but for larger $l$ this may be an issue.
$endgroup$
– астон вілла олоф мэллбэрг
Aug 27 '18 at 12:05
$begingroup$
Do you mean norm of an element, perhaps? (Note that UFDs may not be a Euclidean domain and so there may not be a Euclidean function)
$endgroup$
– daruma
Aug 27 '18 at 13:42
$begingroup$
@daruma, see Field norm
$endgroup$
– lhf
Dec 17 '18 at 11:06
$begingroup$
For $l=5$, see math.stackexchange.com/questions/24840/…. See also math.stackexchange.com/questions/175118/…
$endgroup$
– lhf
Dec 17 '18 at 11:07
$begingroup$
Not sure it is very easy : for a fixed element, would you not need to find the minimal polynomial or the conjugates of that element? For $l=3$ the conjugates were easy to find, but for larger $l$ this may be an issue.
$endgroup$
– астон вілла олоф мэллбэрг
Aug 27 '18 at 12:05
$begingroup$
Not sure it is very easy : for a fixed element, would you not need to find the minimal polynomial or the conjugates of that element? For $l=3$ the conjugates were easy to find, but for larger $l$ this may be an issue.
$endgroup$
– астон вілла олоф мэллбэрг
Aug 27 '18 at 12:05
$begingroup$
Do you mean norm of an element, perhaps? (Note that UFDs may not be a Euclidean domain and so there may not be a Euclidean function)
$endgroup$
– daruma
Aug 27 '18 at 13:42
$begingroup$
Do you mean norm of an element, perhaps? (Note that UFDs may not be a Euclidean domain and so there may not be a Euclidean function)
$endgroup$
– daruma
Aug 27 '18 at 13:42
$begingroup$
@daruma, see Field norm
$endgroup$
– lhf
Dec 17 '18 at 11:06
$begingroup$
@daruma, see Field norm
$endgroup$
– lhf
Dec 17 '18 at 11:06
$begingroup$
For $l=5$, see math.stackexchange.com/questions/24840/…. See also math.stackexchange.com/questions/175118/…
$endgroup$
– lhf
Dec 17 '18 at 11:07
$begingroup$
For $l=5$, see math.stackexchange.com/questions/24840/…. See also math.stackexchange.com/questions/175118/…
$endgroup$
– lhf
Dec 17 '18 at 11:07
add a comment |
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$begingroup$
Not sure it is very easy : for a fixed element, would you not need to find the minimal polynomial or the conjugates of that element? For $l=3$ the conjugates were easy to find, but for larger $l$ this may be an issue.
$endgroup$
– астон вілла олоф мэллбэрг
Aug 27 '18 at 12:05
$begingroup$
Do you mean norm of an element, perhaps? (Note that UFDs may not be a Euclidean domain and so there may not be a Euclidean function)
$endgroup$
– daruma
Aug 27 '18 at 13:42
$begingroup$
@daruma, see Field norm
$endgroup$
– lhf
Dec 17 '18 at 11:06
$begingroup$
For $l=5$, see math.stackexchange.com/questions/24840/…. See also math.stackexchange.com/questions/175118/…
$endgroup$
– lhf
Dec 17 '18 at 11:07