Mixing
Cardinality of a field between interval
$begingroup$
I was studying for an exam when I came across the following question:
Let $F$ a field such that $4<|F|<15$. The number of elements in $F$ is:
A. 6 or 7
B. 11 or 13
C. 8 or 9
D. 7 or 14
For better understanding of the problem, I defined $|x|=|F|, x in mathbb{R}$
Then, after some calculations, I found that $|x|=11$, but I can't understand why there is 2 possible sizes for this field.
Why would the cardinality of $F$ be two different numbers? Is it possible?
field-theory finite-fields
edited Jan 17 at 18:41
SvanN
2,0661422
asked Jan 17 at 18:13
Principiant ForeverPrincipiant Forever
133
$endgroup$
add a comment |
$begingroup$
I was studying for an exam when I came across the following question:
Let $F$ a field such that $4<|F|<15$. The number of elements in $F$ is:
A. 6 or 7
B. 11 or 13
C. 8 or 9
D. 7 or 14
For better understanding of the problem, I defined $|x|=|F|, x in mathbb{R}$
Then, after some calculations, I found that $|x|=11$, but I can't understand why there is 2 possible sizes for this field.
Why would the cardinality of $F$ be two different numbers? Is it possible?
field-theory finite-fields
edited Jan 17 at 18:41
SvanN
2,0661422
asked Jan 17 at 18:13
Principiant ForeverPrincipiant Forever
133
$endgroup$
2
$begingroup$
Both B and C are possible, since a finite field has order a prime power.
$endgroup$
– SvanN
Jan 17 at 18:17
1
$begingroup$
How did you find that $|x|=11$?
$endgroup$
– Servaes
Jan 17 at 19:40
$begingroup$
I used the modulus properties for inequalities. Got to thinking, the answer was more than obvious...
$endgroup$
– Principiant Forever
Jan 18 at 16:14
add a comment |
$begingroup$
I was studying for an exam when I came across the following question:
Let $F$ a field such that $4<|F|<15$. The number of elements in $F$ is:
A. 6 or 7
B. 11 or 13
C. 8 or 9
D. 7 or 14
For better understanding of the problem, I defined $|x|=|F|, x in mathbb{R}$
Then, after some calculations, I found that $|x|=11$, but I can't understand why there is 2 possible sizes for this field.
Why would the cardinality of $F$ be two different numbers? Is it possible?
field-theory finite-fields
edited Jan 17 at 18:41
SvanN
2,0661422
asked Jan 17 at 18:13
Principiant ForeverPrincipiant Forever
133
$endgroup$
I was studying for an exam when I came across the following question:
Let $F$ a field such that $4<|F|<15$. The number of elements in $F$ is:
A. 6 or 7
B. 11 or 13
C. 8 or 9
D. 7 or 14
For better understanding of the problem, I defined $|x|=|F|, x in mathbb{R}$
Then, after some calculations, I found that $|x|=11$, but I can't understand why there is 2 possible sizes for this field.
Why would the cardinality of $F$ be two different numbers? Is it possible?
field-theory finite-fields
field-theory finite-fields
edited Jan 17 at 18:41
SvanN
2,0661422
asked Jan 17 at 18:13
Principiant ForeverPrincipiant Forever
133
edited Jan 17 at 18:41
SvanN
2,0661422
asked Jan 17 at 18:13
Principiant ForeverPrincipiant Forever
133
edited Jan 17 at 18:41
SvanN
2,0661422
edited Jan 17 at 18:41
SvanN
2,0661422
edited Jan 17 at 18:41
SvanN
2,0661422
2,0661422
asked Jan 17 at 18:13
Principiant ForeverPrincipiant Forever
133
asked Jan 17 at 18:13
Principiant ForeverPrincipiant Forever
133
asked Jan 17 at 18:13
Principiant ForeverPrincipiant Forever
133
133
2
$begingroup$
Both B and C are possible, since a finite field has order a prime power.
$endgroup$
– SvanN
Jan 17 at 18:17
1
$begingroup$
How did you find that $|x|=11$?
$endgroup$
– Servaes
Jan 17 at 19:40
$begingroup$
I used the modulus properties for inequalities. Got to thinking, the answer was more than obvious...
$endgroup$
– Principiant Forever
Jan 18 at 16:14
add a comment |
2
$begingroup$
Both B and C are possible, since a finite field has order a prime power.
$endgroup$
– SvanN
Jan 17 at 18:17
1
$begingroup$
How did you find that $|x|=11$?
$endgroup$
– Servaes
Jan 17 at 19:40
$begingroup$
I used the modulus properties for inequalities. Got to thinking, the answer was more than obvious...
$endgroup$
– Principiant Forever
Jan 18 at 16:14
2
2
$begingroup$
Both B and C are possible, since a finite field has order a prime power.
$endgroup$
– SvanN
Jan 17 at 18:17
$begingroup$
Both B and C are possible, since a finite field has order a prime power.
$endgroup$
– SvanN
Jan 17 at 18:17
1
1
$begingroup$
How did you find that $|x|=11$?
$endgroup$
– Servaes
Jan 17 at 19:40
$begingroup$
How did you find that $|x|=11$?
$endgroup$
– Servaes
Jan 17 at 19:40
$begingroup$
I used the modulus properties for inequalities. Got to thinking, the answer was more than obvious...
$endgroup$
– Principiant Forever
Jan 18 at 16:14
$begingroup$
I used the modulus properties for inequalities. Got to thinking, the answer was more than obvious...
$endgroup$
– Principiant Forever
Jan 18 at 16:14
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The question is asking what cardinalities are possible for $F$ to have without any further information being given. The reason for this is that a finite field has as its order the power of some prime number, i.e., $p^n$ for some prime $p$ and $ngeq 1$. The only such numbers between $4$ and $15$ are those in options B and C.
However, all options listed in B and C are possible: there exists a finite field of size $8$, but also one of $9$, or one of $11$, or one of size $13$ (actually, only one for each size, up to isomorphism).
So I think this is a bad question, unless multiple answers were allowed...
answered Jan 17 at 18:42
SvanNSvanN
2,0661422
$endgroup$
3
$begingroup$
I think this is a bad question even if multiple answers were allowed.
$endgroup$
– Servaes
Jan 17 at 19:41
$begingroup$
It helped me understand both the question and the "solution". Thanks!
$endgroup$
– Principiant Forever
Jan 18 at 16:10
add a comment |
$begingroup$
With the limited information given, we cannot conclude the precise order of $F $. A useful hint is that the order of a finite field $F$ is $p^n $ ,where $p$ is the characteristic of $F$(and it's a prime).
answered Jan 17 at 18:21
Thomas ShelbyThomas Shelby
3,5642525
$endgroup$
$begingroup$
"limited information given"? I transcribed the question from an old exam. Maybe it is wrong, or poorly described?
$endgroup$
– Principiant Forever
Jan 17 at 18:31
1
$begingroup$
Please, @PrincipiantForever, never think that there may not be infelicities, or even rank errors in an examination. The people who write examinations are not gods, but all-too-fallible human beings. I shudder to think of all the errors I committed, both large and small, in the exams I wrote when I was teaching.
$endgroup$
– Lubin
Jan 18 at 0:24
$begingroup$
@PrincipiantForever The question is poorly described.
$endgroup$
– Thomas Shelby
Jan 18 at 1:08
$begingroup$
@Lubin you can't be more right about it. Unfortunately, I believe this causes great impact on students lives.
$endgroup$
– Principiant Forever
Jan 18 at 16:16
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The question is asking what cardinalities are possible for $F$ to have without any further information being given. The reason for this is that a finite field has as its order the power of some prime number, i.e., $p^n$ for some prime $p$ and $ngeq 1$. The only such numbers between $4$ and $15$ are those in options B and C.
However, all options listed in B and C are possible: there exists a finite field of size $8$, but also one of $9$, or one of $11$, or one of size $13$ (actually, only one for each size, up to isomorphism).
So I think this is a bad question, unless multiple answers were allowed...
answered Jan 17 at 18:42
SvanNSvanN
2,0661422
$endgroup$
3
$begingroup$
I think this is a bad question even if multiple answers were allowed.
$endgroup$
– Servaes
Jan 17 at 19:41
$begingroup$
It helped me understand both the question and the "solution". Thanks!
$endgroup$
– Principiant Forever
Jan 18 at 16:10
add a comment |
$begingroup$
The question is asking what cardinalities are possible for $F$ to have without any further information being given. The reason for this is that a finite field has as its order the power of some prime number, i.e., $p^n$ for some prime $p$ and $ngeq 1$. The only such numbers between $4$ and $15$ are those in options B and C.
However, all options listed in B and C are possible: there exists a finite field of size $8$, but also one of $9$, or one of $11$, or one of size $13$ (actually, only one for each size, up to isomorphism).
So I think this is a bad question, unless multiple answers were allowed...
answered Jan 17 at 18:42
SvanNSvanN
2,0661422
$endgroup$
3
$begingroup$
I think this is a bad question even if multiple answers were allowed.
$endgroup$
– Servaes
Jan 17 at 19:41
$begingroup$
It helped me understand both the question and the "solution". Thanks!
$endgroup$
– Principiant Forever
Jan 18 at 16:10
add a comment |
$begingroup$
The question is asking what cardinalities are possible for $F$ to have without any further information being given. The reason for this is that a finite field has as its order the power of some prime number, i.e., $p^n$ for some prime $p$ and $ngeq 1$. The only such numbers between $4$ and $15$ are those in options B and C.
However, all options listed in B and C are possible: there exists a finite field of size $8$, but also one of $9$, or one of $11$, or one of size $13$ (actually, only one for each size, up to isomorphism).
So I think this is a bad question, unless multiple answers were allowed...
answered Jan 17 at 18:42
SvanNSvanN
2,0661422
$endgroup$
The question is asking what cardinalities are possible for $F$ to have without any further information being given. The reason for this is that a finite field has as its order the power of some prime number, i.e., $p^n$ for some prime $p$ and $ngeq 1$. The only such numbers between $4$ and $15$ are those in options B and C.
However, all options listed in B and C are possible: there exists a finite field of size $8$, but also one of $9$, or one of $11$, or one of size $13$ (actually, only one for each size, up to isomorphism).
So I think this is a bad question, unless multiple answers were allowed...
answered Jan 17 at 18:42
SvanNSvanN
2,0661422
answered Jan 17 at 18:42
SvanNSvanN
2,0661422
answered Jan 17 at 18:42
SvanNSvanN
2,0661422
answered Jan 17 at 18:42
SvanNSvanN
2,0661422
2,0661422
3
$begingroup$
I think this is a bad question even if multiple answers were allowed.
$endgroup$
– Servaes
Jan 17 at 19:41
$begingroup$
It helped me understand both the question and the "solution". Thanks!
$endgroup$
– Principiant Forever
Jan 18 at 16:10
add a comment |
3
$begingroup$
I think this is a bad question even if multiple answers were allowed.
$endgroup$
– Servaes
Jan 17 at 19:41
$begingroup$
It helped me understand both the question and the "solution". Thanks!
$endgroup$
– Principiant Forever
Jan 18 at 16:10
3
3
$begingroup$
I think this is a bad question even if multiple answers were allowed.
$endgroup$
– Servaes
Jan 17 at 19:41
$begingroup$
I think this is a bad question even if multiple answers were allowed.
$endgroup$
– Servaes
Jan 17 at 19:41
$begingroup$
It helped me understand both the question and the "solution". Thanks!
$endgroup$
– Principiant Forever
Jan 18 at 16:10
$begingroup$
It helped me understand both the question and the "solution". Thanks!
$endgroup$
– Principiant Forever
Jan 18 at 16:10
add a comment |
$begingroup$
With the limited information given, we cannot conclude the precise order of $F $. A useful hint is that the order of a finite field $F$ is $p^n $ ,where $p$ is the characteristic of $F$(and it's a prime).
answered Jan 17 at 18:21
Thomas ShelbyThomas Shelby
3,5642525
$endgroup$
$begingroup$
"limited information given"? I transcribed the question from an old exam. Maybe it is wrong, or poorly described?
$endgroup$
– Principiant Forever
Jan 17 at 18:31
1
$begingroup$
Please, @PrincipiantForever, never think that there may not be infelicities, or even rank errors in an examination. The people who write examinations are not gods, but all-too-fallible human beings. I shudder to think of all the errors I committed, both large and small, in the exams I wrote when I was teaching.
$endgroup$
– Lubin
Jan 18 at 0:24
$begingroup$
@PrincipiantForever The question is poorly described.
$endgroup$
– Thomas Shelby
Jan 18 at 1:08
$begingroup$
@Lubin you can't be more right about it. Unfortunately, I believe this causes great impact on students lives.
$endgroup$
– Principiant Forever
Jan 18 at 16:16
add a comment |
$begingroup$
With the limited information given, we cannot conclude the precise order of $F $. A useful hint is that the order of a finite field $F$ is $p^n $ ,where $p$ is the characteristic of $F$(and it's a prime).
answered Jan 17 at 18:21
Thomas ShelbyThomas Shelby
3,5642525
$endgroup$
$begingroup$
"limited information given"? I transcribed the question from an old exam. Maybe it is wrong, or poorly described?
$endgroup$
– Principiant Forever
Jan 17 at 18:31
1
$begingroup$
Please, @PrincipiantForever, never think that there may not be infelicities, or even rank errors in an examination. The people who write examinations are not gods, but all-too-fallible human beings. I shudder to think of all the errors I committed, both large and small, in the exams I wrote when I was teaching.
$endgroup$
– Lubin
Jan 18 at 0:24
$begingroup$
@PrincipiantForever The question is poorly described.
$endgroup$
– Thomas Shelby
Jan 18 at 1:08
$begingroup$
@Lubin you can't be more right about it. Unfortunately, I believe this causes great impact on students lives.
$endgroup$
– Principiant Forever
Jan 18 at 16:16
add a comment |
$begingroup$
With the limited information given, we cannot conclude the precise order of $F $. A useful hint is that the order of a finite field $F$ is $p^n $ ,where $p$ is the characteristic of $F$(and it's a prime).
answered Jan 17 at 18:21
Thomas ShelbyThomas Shelby
3,5642525
$endgroup$
With the limited information given, we cannot conclude the precise order of $F $. A useful hint is that the order of a finite field $F$ is $p^n $ ,where $p$ is the characteristic of $F$(and it's a prime).
answered Jan 17 at 18:21
Thomas ShelbyThomas Shelby
3,5642525
edited Jan 18 at 1:40
answered Jan 17 at 18:21
Thomas ShelbyThomas Shelby
3,5642525
answered Jan 17 at 18:21
Thomas ShelbyThomas Shelby
3,5642525
answered Jan 17 at 18:21
Thomas ShelbyThomas Shelby
3,5642525
3,5642525
$begingroup$
"limited information given"? I transcribed the question from an old exam. Maybe it is wrong, or poorly described?
$endgroup$
– Principiant Forever
Jan 17 at 18:31
1
$begingroup$
Please, @PrincipiantForever, never think that there may not be infelicities, or even rank errors in an examination. The people who write examinations are not gods, but all-too-fallible human beings. I shudder to think of all the errors I committed, both large and small, in the exams I wrote when I was teaching.
$endgroup$
– Lubin
Jan 18 at 0:24
$begingroup$
@PrincipiantForever The question is poorly described.
$endgroup$
– Thomas Shelby
Jan 18 at 1:08
$begingroup$
@Lubin you can't be more right about it. Unfortunately, I believe this causes great impact on students lives.
$endgroup$
– Principiant Forever
Jan 18 at 16:16
add a comment |
$begingroup$
"limited information given"? I transcribed the question from an old exam. Maybe it is wrong, or poorly described?
$endgroup$
– Principiant Forever
Jan 17 at 18:31
1
$begingroup$
Please, @PrincipiantForever, never think that there may not be infelicities, or even rank errors in an examination. The people who write examinations are not gods, but all-too-fallible human beings. I shudder to think of all the errors I committed, both large and small, in the exams I wrote when I was teaching.
$endgroup$
– Lubin
Jan 18 at 0:24
$begingroup$
@PrincipiantForever The question is poorly described.
$endgroup$
– Thomas Shelby
Jan 18 at 1:08
$begingroup$
@Lubin you can't be more right about it. Unfortunately, I believe this causes great impact on students lives.
$endgroup$
– Principiant Forever
Jan 18 at 16:16
$begingroup$
"limited information given"? I transcribed the question from an old exam. Maybe it is wrong, or poorly described?
$endgroup$
– Principiant Forever
Jan 17 at 18:31
$begingroup$
"limited information given"? I transcribed the question from an old exam. Maybe it is wrong, or poorly described?
$endgroup$
– Principiant Forever
Jan 17 at 18:31
1
1
$begingroup$
Please, @PrincipiantForever, never think that there may not be infelicities, or even rank errors in an examination. The people who write examinations are not gods, but all-too-fallible human beings. I shudder to think of all the errors I committed, both large and small, in the exams I wrote when I was teaching.
$endgroup$
– Lubin
Jan 18 at 0:24
$begingroup$
Please, @PrincipiantForever, never think that there may not be infelicities, or even rank errors in an examination. The people who write examinations are not gods, but all-too-fallible human beings. I shudder to think of all the errors I committed, both large and small, in the exams I wrote when I was teaching.
$endgroup$
– Lubin
Jan 18 at 0:24
$begingroup$
@PrincipiantForever The question is poorly described.
$endgroup$
– Thomas Shelby
Jan 18 at 1:08
$begingroup$
@PrincipiantForever The question is poorly described.
$endgroup$
– Thomas Shelby
Jan 18 at 1:08
$begingroup$
@Lubin you can't be more right about it. Unfortunately, I believe this causes great impact on students lives.
$endgroup$
– Principiant Forever
Jan 18 at 16:16
$begingroup$
@Lubin you can't be more right about it. Unfortunately, I believe this causes great impact on students lives.
$endgroup$
– Principiant Forever
Jan 18 at 16:16
add a comment |
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2
$begingroup$
Both B and C are possible, since a finite field has order a prime power.
$endgroup$
– SvanN
Jan 17 at 18:17
1
$begingroup$
How did you find that $|x|=11$?
$endgroup$
– Servaes
Jan 17 at 19:40
$begingroup$
I used the modulus properties for inequalities. Got to thinking, the answer was more than obvious...
$endgroup$
– Principiant Forever
Jan 18 at 16:14