Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? [closed]
$text{p-adic numbers}:$
My questions are-
$(1)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty?
$(2)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p $ non-empty?
$(3)$ If non-empty , then what are the intersection sets ?
I can not conclude the answer.
Please someone help me with details answer or at least hints.
number-theory p-adic-number-theory
asked Nov 28 '18 at 11:17
M. A. SARKAR
2,1831619
closed as too broad by Saad, Watson, paul garrett, user10354138, José Carlos Santos Nov 28 '18 at 15:37
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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$text{p-adic numbers}:$
My questions are-
$(1)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty?
$(2)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p $ non-empty?
$(3)$ If non-empty , then what are the intersection sets ?
I can not conclude the answer.
Please someone help me with details answer or at least hints.
number-theory p-adic-number-theory
asked Nov 28 '18 at 11:17
M. A. SARKAR
2,1831619
closed as too broad by Saad, Watson, paul garrett, user10354138, José Carlos Santos Nov 28 '18 at 15:37
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$text{p-adic numbers}:$
My questions are-
$(1)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty?
$(2)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p $ non-empty?
$(3)$ If non-empty , then what are the intersection sets ?
I can not conclude the answer.
Please someone help me with details answer or at least hints.
number-theory p-adic-number-theory
asked Nov 28 '18 at 11:17
M. A. SARKAR
2,1831619
$text{p-adic numbers}:$
My questions are-
$(1)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty?
$(2)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p $ non-empty?
$(3)$ If non-empty , then what are the intersection sets ?
I can not conclude the answer.
Please someone help me with details answer or at least hints.
number-theory p-adic-number-theory
number-theory p-adic-number-theory
asked Nov 28 '18 at 11:17
M. A. SARKAR
2,1831619
asked Nov 28 '18 at 11:17
M. A. SARKAR
2,1831619
edited Nov 28 '18 at 16:54
asked Nov 28 '18 at 11:17
M. A. SARKAR
2,1831619
asked Nov 28 '18 at 11:17
M. A. SARKAR
2,1831619
asked Nov 28 '18 at 11:17
M. A. SARKAR
2,1831619
2,1831619
closed as too broad by Saad, Watson, paul garrett, user10354138, José Carlos Santos Nov 28 '18 at 15:37
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as too broad by Saad, Watson, paul garrett, user10354138, José Carlos Santos Nov 28 '18 at 15:37
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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1 Answer
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If $q$ is a prime distinct from $p$, then $1/qinmathbb{Z}_{p}capmathbb{Q}$ and $1/qnotinmathbb{Z}$. Thus $1/q$ belongs to the set in $(1)$.
Since $mathbb{Z}_psetminusmathbb{Z}subseteqmathbb{Q}_{p}$, the set in $(2)$ is the same as $mathbb{Z}_psetminusmathbb{Z}$.
answered Nov 28 '18 at 13:43
egreg
178k1484201
Please answer part $(1)$ or $(2) $ keeping in mind of part $(3)$ question. I asked in part $(3)$ if $ (mathbb{Z}_p setminus mathbb{Z} ) cap mathbb{Q} $ or $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p$ are non-empty then which elements belong to it or what would be the sets? Please help me
– M. A. SARKAR
Nov 28 '18 at 14:13
Please reply my comment above . Also in your answer you said that $ frac{1}{q} $ belongs to the set in $(1)$ if $ q$ is a prime different from $p$. . My question is why only primes, why not any $ frac{1}{n}$ belong to the set in $(1)$?
– M. A. SARKAR
Nov 29 '18 at 4:15
1
@M.A.SARKAR That's a simple proof that the set is not empty. I never said that just elements of the form $1/q$ are there. Actually, the set in (1) is exactly $mathbb{Z}_{(p)}setminusmathbb{Z}$.
– egreg
Nov 29 '18 at 9:56
I got it after all
– M. A. SARKAR
Nov 29 '18 at 10:30
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
If $q$ is a prime distinct from $p$, then $1/qinmathbb{Z}_{p}capmathbb{Q}$ and $1/qnotinmathbb{Z}$. Thus $1/q$ belongs to the set in $(1)$.
Since $mathbb{Z}_psetminusmathbb{Z}subseteqmathbb{Q}_{p}$, the set in $(2)$ is the same as $mathbb{Z}_psetminusmathbb{Z}$.
answered Nov 28 '18 at 13:43
egreg
178k1484201
Please answer part $(1)$ or $(2) $ keeping in mind of part $(3)$ question. I asked in part $(3)$ if $ (mathbb{Z}_p setminus mathbb{Z} ) cap mathbb{Q} $ or $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p$ are non-empty then which elements belong to it or what would be the sets? Please help me
– M. A. SARKAR
Nov 28 '18 at 14:13
Please reply my comment above . Also in your answer you said that $ frac{1}{q} $ belongs to the set in $(1)$ if $ q$ is a prime different from $p$. . My question is why only primes, why not any $ frac{1}{n}$ belong to the set in $(1)$?
– M. A. SARKAR
Nov 29 '18 at 4:15
1
@M.A.SARKAR That's a simple proof that the set is not empty. I never said that just elements of the form $1/q$ are there. Actually, the set in (1) is exactly $mathbb{Z}_{(p)}setminusmathbb{Z}$.
– egreg
Nov 29 '18 at 9:56
I got it after all
– M. A. SARKAR
Nov 29 '18 at 10:30
add a comment |
If $q$ is a prime distinct from $p$, then $1/qinmathbb{Z}_{p}capmathbb{Q}$ and $1/qnotinmathbb{Z}$. Thus $1/q$ belongs to the set in $(1)$.
Since $mathbb{Z}_psetminusmathbb{Z}subseteqmathbb{Q}_{p}$, the set in $(2)$ is the same as $mathbb{Z}_psetminusmathbb{Z}$.
answered Nov 28 '18 at 13:43
egreg
178k1484201
Please answer part $(1)$ or $(2) $ keeping in mind of part $(3)$ question. I asked in part $(3)$ if $ (mathbb{Z}_p setminus mathbb{Z} ) cap mathbb{Q} $ or $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p$ are non-empty then which elements belong to it or what would be the sets? Please help me
– M. A. SARKAR
Nov 28 '18 at 14:13
Please reply my comment above . Also in your answer you said that $ frac{1}{q} $ belongs to the set in $(1)$ if $ q$ is a prime different from $p$. . My question is why only primes, why not any $ frac{1}{n}$ belong to the set in $(1)$?
– M. A. SARKAR
Nov 29 '18 at 4:15
1
@M.A.SARKAR That's a simple proof that the set is not empty. I never said that just elements of the form $1/q$ are there. Actually, the set in (1) is exactly $mathbb{Z}_{(p)}setminusmathbb{Z}$.
– egreg
Nov 29 '18 at 9:56
I got it after all
– M. A. SARKAR
Nov 29 '18 at 10:30
add a comment |
If $q$ is a prime distinct from $p$, then $1/qinmathbb{Z}_{p}capmathbb{Q}$ and $1/qnotinmathbb{Z}$. Thus $1/q$ belongs to the set in $(1)$.
Since $mathbb{Z}_psetminusmathbb{Z}subseteqmathbb{Q}_{p}$, the set in $(2)$ is the same as $mathbb{Z}_psetminusmathbb{Z}$.
answered Nov 28 '18 at 13:43
egreg
178k1484201
If $q$ is a prime distinct from $p$, then $1/qinmathbb{Z}_{p}capmathbb{Q}$ and $1/qnotinmathbb{Z}$. Thus $1/q$ belongs to the set in $(1)$.
Since $mathbb{Z}_psetminusmathbb{Z}subseteqmathbb{Q}_{p}$, the set in $(2)$ is the same as $mathbb{Z}_psetminusmathbb{Z}$.
answered Nov 28 '18 at 13:43
egreg
178k1484201
answered Nov 28 '18 at 13:43
egreg
178k1484201
answered Nov 28 '18 at 13:43
egreg
178k1484201
answered Nov 28 '18 at 13:43
egreg
178k1484201
178k1484201
Please answer part $(1)$ or $(2) $ keeping in mind of part $(3)$ question. I asked in part $(3)$ if $ (mathbb{Z}_p setminus mathbb{Z} ) cap mathbb{Q} $ or $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p$ are non-empty then which elements belong to it or what would be the sets? Please help me
– M. A. SARKAR
Nov 28 '18 at 14:13
Please reply my comment above . Also in your answer you said that $ frac{1}{q} $ belongs to the set in $(1)$ if $ q$ is a prime different from $p$. . My question is why only primes, why not any $ frac{1}{n}$ belong to the set in $(1)$?
– M. A. SARKAR
Nov 29 '18 at 4:15
1
@M.A.SARKAR That's a simple proof that the set is not empty. I never said that just elements of the form $1/q$ are there. Actually, the set in (1) is exactly $mathbb{Z}_{(p)}setminusmathbb{Z}$.
– egreg
Nov 29 '18 at 9:56
I got it after all
– M. A. SARKAR
Nov 29 '18 at 10:30
add a comment |
Please answer part $(1)$ or $(2) $ keeping in mind of part $(3)$ question. I asked in part $(3)$ if $ (mathbb{Z}_p setminus mathbb{Z} ) cap mathbb{Q} $ or $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p$ are non-empty then which elements belong to it or what would be the sets? Please help me
– M. A. SARKAR
Nov 28 '18 at 14:13
Please reply my comment above . Also in your answer you said that $ frac{1}{q} $ belongs to the set in $(1)$ if $ q$ is a prime different from $p$. . My question is why only primes, why not any $ frac{1}{n}$ belong to the set in $(1)$?
– M. A. SARKAR
Nov 29 '18 at 4:15
1
@M.A.SARKAR That's a simple proof that the set is not empty. I never said that just elements of the form $1/q$ are there. Actually, the set in (1) is exactly $mathbb{Z}_{(p)}setminusmathbb{Z}$.
– egreg
Nov 29 '18 at 9:56
I got it after all
– M. A. SARKAR
Nov 29 '18 at 10:30
Please answer part $(1)$ or $(2) $ keeping in mind of part $(3)$ question. I asked in part $(3)$ if $ (mathbb{Z}_p setminus mathbb{Z} ) cap mathbb{Q} $ or $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p$ are non-empty then which elements belong to it or what would be the sets? Please help me
– M. A. SARKAR
Nov 28 '18 at 14:13
Please answer part $(1)$ or $(2) $ keeping in mind of part $(3)$ question. I asked in part $(3)$ if $ (mathbb{Z}_p setminus mathbb{Z} ) cap mathbb{Q} $ or $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p$ are non-empty then which elements belong to it or what would be the sets? Please help me
– M. A. SARKAR
Nov 28 '18 at 14:13
Please reply my comment above . Also in your answer you said that $ frac{1}{q} $ belongs to the set in $(1)$ if $ q$ is a prime different from $p$. . My question is why only primes, why not any $ frac{1}{n}$ belong to the set in $(1)$?
– M. A. SARKAR
Nov 29 '18 at 4:15
Please reply my comment above . Also in your answer you said that $ frac{1}{q} $ belongs to the set in $(1)$ if $ q$ is a prime different from $p$. . My question is why only primes, why not any $ frac{1}{n}$ belong to the set in $(1)$?
– M. A. SARKAR
Nov 29 '18 at 4:15
1
1
@M.A.SARKAR That's a simple proof that the set is not empty. I never said that just elements of the form $1/q$ are there. Actually, the set in (1) is exactly $mathbb{Z}_{(p)}setminusmathbb{Z}$.
– egreg
Nov 29 '18 at 9:56
@M.A.SARKAR That's a simple proof that the set is not empty. I never said that just elements of the form $1/q$ are there. Actually, the set in (1) is exactly $mathbb{Z}_{(p)}setminusmathbb{Z}$.
– egreg
Nov 29 '18 at 9:56
I got it after all
– M. A. SARKAR
Nov 29 '18 at 10:30
I got it after all
– M. A. SARKAR
Nov 29 '18 at 10:30
add a comment |