Mixing
Why is GCD of any two integers positive? [unit normalization of GCDs]
The definition in my text reads,
An integer $d$ is said to be the greatest common divisor of two non-zero integers $a$ and $b$ iff,
$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$
then $k|d$
Now here's the thing, if $d|a$ and $d|b$ then surely $-d|a$ and $-d|b$ as well, also $k|-d$
What I take from this? GCD is not unique! That is if $mathrm{gcd}(12,8)= 4$ then by the definition, $mathrm{gcd}(12,8) = -4$ as well.
Yet I never ever seen a negative gcd. Someone please explain.
Maybe, $4>-4$, and we want the "greatest common factor" so...? But that still doesn't justifies the definition.
abstract-algebra elementary-number-theory greatest-common-divisor
edited Nov 21 '18 at 2:36
Bill Dubuque
209k29190629
asked Oct 2 '18 at 10:12
William
1,170314
|
show 2 more comments
The definition in my text reads,
An integer $d$ is said to be the greatest common divisor of two non-zero integers $a$ and $b$ iff,
$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$
then $k|d$
Now here's the thing, if $d|a$ and $d|b$ then surely $-d|a$ and $-d|b$ as well, also $k|-d$
What I take from this? GCD is not unique! That is if $mathrm{gcd}(12,8)= 4$ then by the definition, $mathrm{gcd}(12,8) = -4$ as well.
Yet I never ever seen a negative gcd. Someone please explain.
Maybe, $4>-4$, and we want the "greatest common factor" so...? But that still doesn't justifies the definition.
abstract-algebra elementary-number-theory greatest-common-divisor
edited Nov 21 '18 at 2:36
Bill Dubuque
209k29190629
asked Oct 2 '18 at 10:12
William
1,170314
2
Where is than definition from? From wikipedia: "In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers"
– Roberto Rastapopoulos
Oct 2 '18 at 10:15
I expect your definition required $din mathbb N$,
– lulu
Oct 2 '18 at 10:18
3
Indeed, the greatest common divisor of two elements is only uniquely determined up to an invertible element (i.e. up to a sign in the integers). For the integers we usually choose the positive one, since this is a "nice" way to single out one of them.
– Tobias Kildetoft
Oct 2 '18 at 10:20
@RobertoRastapopoulos you can find it many algebra textbooks, but here is an example in the form of question. math.stackexchange.com/q/85565/552998
– William
Oct 2 '18 at 10:24
2
In some definitions it's a gcd (because there can be more than one)
– John Cataldo
Oct 2 '18 at 10:42
|
show 2 more comments
The definition in my text reads,
An integer $d$ is said to be the greatest common divisor of two non-zero integers $a$ and $b$ iff,
$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$
then $k|d$
Now here's the thing, if $d|a$ and $d|b$ then surely $-d|a$ and $-d|b$ as well, also $k|-d$
What I take from this? GCD is not unique! That is if $mathrm{gcd}(12,8)= 4$ then by the definition, $mathrm{gcd}(12,8) = -4$ as well.
Yet I never ever seen a negative gcd. Someone please explain.
Maybe, $4>-4$, and we want the "greatest common factor" so...? But that still doesn't justifies the definition.
abstract-algebra elementary-number-theory greatest-common-divisor
edited Nov 21 '18 at 2:36
Bill Dubuque
209k29190629
asked Oct 2 '18 at 10:12
William
1,170314
The definition in my text reads,
An integer $d$ is said to be the greatest common divisor of two non-zero integers $a$ and $b$ iff,
$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$
then $k|d$
Now here's the thing, if $d|a$ and $d|b$ then surely $-d|a$ and $-d|b$ as well, also $k|-d$
What I take from this? GCD is not unique! That is if $mathrm{gcd}(12,8)= 4$ then by the definition, $mathrm{gcd}(12,8) = -4$ as well.
Yet I never ever seen a negative gcd. Someone please explain.
Maybe, $4>-4$, and we want the "greatest common factor" so...? But that still doesn't justifies the definition.
abstract-algebra elementary-number-theory greatest-common-divisor
abstract-algebra elementary-number-theory greatest-common-divisor
edited Nov 21 '18 at 2:36
Bill Dubuque
209k29190629
asked Oct 2 '18 at 10:12
William
1,170314
edited Nov 21 '18 at 2:36
Bill Dubuque
209k29190629
asked Oct 2 '18 at 10:12
William
1,170314
edited Nov 21 '18 at 2:36
Bill Dubuque
209k29190629
edited Nov 21 '18 at 2:36
Bill Dubuque
209k29190629
edited Nov 21 '18 at 2:36
Bill Dubuque
209k29190629
209k29190629
asked Oct 2 '18 at 10:12
William
1,170314
asked Oct 2 '18 at 10:12
William
1,170314
asked Oct 2 '18 at 10:12
William
1,170314
1,170314
2
Where is than definition from? From wikipedia: "In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers"
– Roberto Rastapopoulos
Oct 2 '18 at 10:15
I expect your definition required $din mathbb N$,
– lulu
Oct 2 '18 at 10:18
3
Indeed, the greatest common divisor of two elements is only uniquely determined up to an invertible element (i.e. up to a sign in the integers). For the integers we usually choose the positive one, since this is a "nice" way to single out one of them.
– Tobias Kildetoft
Oct 2 '18 at 10:20
@RobertoRastapopoulos you can find it many algebra textbooks, but here is an example in the form of question. math.stackexchange.com/q/85565/552998
– William
Oct 2 '18 at 10:24
2
In some definitions it's a gcd (because there can be more than one)
– John Cataldo
Oct 2 '18 at 10:42
|
show 2 more comments
2
Where is than definition from? From wikipedia: "In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers"
– Roberto Rastapopoulos
Oct 2 '18 at 10:15
I expect your definition required $din mathbb N$,
– lulu
Oct 2 '18 at 10:18
3
Indeed, the greatest common divisor of two elements is only uniquely determined up to an invertible element (i.e. up to a sign in the integers). For the integers we usually choose the positive one, since this is a "nice" way to single out one of them.
– Tobias Kildetoft
Oct 2 '18 at 10:20
@RobertoRastapopoulos you can find it many algebra textbooks, but here is an example in the form of question. math.stackexchange.com/q/85565/552998
– William
Oct 2 '18 at 10:24
2
In some definitions it's a gcd (because there can be more than one)
– John Cataldo
Oct 2 '18 at 10:42
2
2
Where is than definition from? From wikipedia: "In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers"
– Roberto Rastapopoulos
Oct 2 '18 at 10:15
Where is than definition from? From wikipedia: "In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers"
– Roberto Rastapopoulos
Oct 2 '18 at 10:15
I expect your definition required $din mathbb N$,
– lulu
Oct 2 '18 at 10:18
I expect your definition required $din mathbb N$,
– lulu
Oct 2 '18 at 10:18
3
3
Indeed, the greatest common divisor of two elements is only uniquely determined up to an invertible element (i.e. up to a sign in the integers). For the integers we usually choose the positive one, since this is a "nice" way to single out one of them.
– Tobias Kildetoft
Oct 2 '18 at 10:20
Indeed, the greatest common divisor of two elements is only uniquely determined up to an invertible element (i.e. up to a sign in the integers). For the integers we usually choose the positive one, since this is a "nice" way to single out one of them.
– Tobias Kildetoft
Oct 2 '18 at 10:20
@RobertoRastapopoulos you can find it many algebra textbooks, but here is an example in the form of question. math.stackexchange.com/q/85565/552998
– William
Oct 2 '18 at 10:24
@RobertoRastapopoulos you can find it many algebra textbooks, but here is an example in the form of question. math.stackexchange.com/q/85565/552998
– William
Oct 2 '18 at 10:24
2
2
In some definitions it's a gcd (because there can be more than one)
– John Cataldo
Oct 2 '18 at 10:42
In some definitions it's a gcd (because there can be more than one)
– John Cataldo
Oct 2 '18 at 10:42
|
show 2 more comments
2 Answers
2
active
oldest
votes
You are right, it is all about definitions. With the definition you gave the gcd is really not unique and it might be negative as well. But because most of the time we are using only the positive gcd then some simply prefer to add the words "$d$ is positive" to your definition or just give other definitions. For example a very common definition of gcd in number theory is "$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$ then $kleq d$". So that definition already requires the gcd to be positive.
answered Oct 2 '18 at 10:20
Mark
6,050415
But then again, if you agree that gcd can be positive as well as negative, obviously, positive > negative. So the positive divisor should be considered as it's greater. How can gcd be negative in the first place, as it asks for the "greatest" factor.
– William
Oct 2 '18 at 12:31
1
So again, it's up to how you define. The definition I gave you obviously requires the gcd to be positive and hence unique. But in other places you might find a definition that doesn't require that and allows the gcd to be negative as well. (your definition for example). So all the question is do you call the negative number a gcd. Some people do, some do not. The fact is we usually don't really need the negative gcd so in most books you will see a definition according to which the gcd must be positive.
– Mark
Oct 2 '18 at 12:37
Thank you :) I'm just having a hard time accepting that a particular concept has 2 definition, both of them are slightly different from one another. This just doesn't seem math-likely though...
– William
Oct 2 '18 at 14:55
Well, note that all the definitions of the positive gcd are equivalent, there are no contradictions here. It's just that question-do you want to call the negative number a gcd or not? It's all just about names. I prefer the gcd to be positive and unique, it's much easier to think this way.
– Mark
Oct 2 '18 at 16:40
add a comment |
The text is using the universal definition of a gcd, namely
$$ cmid a,b iff cmid gcd(a,b)$$
Direction $(Leftarrow)$ implies that gcd is a common divisor of $a,b,$ (by choosing $ c = gcd(a,b))$ and the reverse direction $(Rightarrow)$ implies that the gcd is "greatest" w.r.t. divisibility order, i.e. divisible by all other common divisors $c$ of $a,b$.
Scaling the gcd by a unit (invertible) $u$ (e.g. $-1)$ preserves the above definition, so generally gcds are unique only up to unit multiples, i.e. up to associates. In some rings with simple unit group structure we can choose canonical representatives of associate classes, which allows is to choose normal-forms for gcds, e.g. in $,Bbb Z,$ (with units $pm 1)$ we normalize gcds $ge 0,,$ and in a polynomial ring $,K[x],$ over a field (units = constants $0neq cin K) $ we normalize polynomial gcds to be monic (lead coeff $,c_n = 1),,$ by scaling the polynomial by $,c_n^{-1},$ if need be (so a constant gcd $,c_0neq 0$ normalizes to $1).,$ Hence in both cases we can say that two elements are coprime if their gcd $= 1$ (vs. a unit). Such normalizations are sometimes called unit normal representatives in the literature.
answered Oct 2 '18 at 16:00
Bill Dubuque
209k29190629
add a comment |
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2 Answers
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2 Answers
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You are right, it is all about definitions. With the definition you gave the gcd is really not unique and it might be negative as well. But because most of the time we are using only the positive gcd then some simply prefer to add the words "$d$ is positive" to your definition or just give other definitions. For example a very common definition of gcd in number theory is "$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$ then $kleq d$". So that definition already requires the gcd to be positive.
answered Oct 2 '18 at 10:20
Mark
6,050415
But then again, if you agree that gcd can be positive as well as negative, obviously, positive > negative. So the positive divisor should be considered as it's greater. How can gcd be negative in the first place, as it asks for the "greatest" factor.
– William
Oct 2 '18 at 12:31
1
So again, it's up to how you define. The definition I gave you obviously requires the gcd to be positive and hence unique. But in other places you might find a definition that doesn't require that and allows the gcd to be negative as well. (your definition for example). So all the question is do you call the negative number a gcd. Some people do, some do not. The fact is we usually don't really need the negative gcd so in most books you will see a definition according to which the gcd must be positive.
– Mark
Oct 2 '18 at 12:37
Thank you :) I'm just having a hard time accepting that a particular concept has 2 definition, both of them are slightly different from one another. This just doesn't seem math-likely though...
– William
Oct 2 '18 at 14:55
Well, note that all the definitions of the positive gcd are equivalent, there are no contradictions here. It's just that question-do you want to call the negative number a gcd or not? It's all just about names. I prefer the gcd to be positive and unique, it's much easier to think this way.
– Mark
Oct 2 '18 at 16:40
add a comment |
You are right, it is all about definitions. With the definition you gave the gcd is really not unique and it might be negative as well. But because most of the time we are using only the positive gcd then some simply prefer to add the words "$d$ is positive" to your definition or just give other definitions. For example a very common definition of gcd in number theory is "$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$ then $kleq d$". So that definition already requires the gcd to be positive.
answered Oct 2 '18 at 10:20
Mark
6,050415
But then again, if you agree that gcd can be positive as well as negative, obviously, positive > negative. So the positive divisor should be considered as it's greater. How can gcd be negative in the first place, as it asks for the "greatest" factor.
– William
Oct 2 '18 at 12:31
1
So again, it's up to how you define. The definition I gave you obviously requires the gcd to be positive and hence unique. But in other places you might find a definition that doesn't require that and allows the gcd to be negative as well. (your definition for example). So all the question is do you call the negative number a gcd. Some people do, some do not. The fact is we usually don't really need the negative gcd so in most books you will see a definition according to which the gcd must be positive.
– Mark
Oct 2 '18 at 12:37
Thank you :) I'm just having a hard time accepting that a particular concept has 2 definition, both of them are slightly different from one another. This just doesn't seem math-likely though...
– William
Oct 2 '18 at 14:55
Well, note that all the definitions of the positive gcd are equivalent, there are no contradictions here. It's just that question-do you want to call the negative number a gcd or not? It's all just about names. I prefer the gcd to be positive and unique, it's much easier to think this way.
– Mark
Oct 2 '18 at 16:40
add a comment |
You are right, it is all about definitions. With the definition you gave the gcd is really not unique and it might be negative as well. But because most of the time we are using only the positive gcd then some simply prefer to add the words "$d$ is positive" to your definition or just give other definitions. For example a very common definition of gcd in number theory is "$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$ then $kleq d$". So that definition already requires the gcd to be positive.
answered Oct 2 '18 at 10:20
Mark
6,050415
You are right, it is all about definitions. With the definition you gave the gcd is really not unique and it might be negative as well. But because most of the time we are using only the positive gcd then some simply prefer to add the words "$d$ is positive" to your definition or just give other definitions. For example a very common definition of gcd in number theory is "$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$ then $kleq d$". So that definition already requires the gcd to be positive.
answered Oct 2 '18 at 10:20
Mark
6,050415
answered Oct 2 '18 at 10:20
Mark
6,050415
answered Oct 2 '18 at 10:20
Mark
6,050415
answered Oct 2 '18 at 10:20
Mark
6,050415
6,050415
But then again, if you agree that gcd can be positive as well as negative, obviously, positive > negative. So the positive divisor should be considered as it's greater. How can gcd be negative in the first place, as it asks for the "greatest" factor.
– William
Oct 2 '18 at 12:31
1
So again, it's up to how you define. The definition I gave you obviously requires the gcd to be positive and hence unique. But in other places you might find a definition that doesn't require that and allows the gcd to be negative as well. (your definition for example). So all the question is do you call the negative number a gcd. Some people do, some do not. The fact is we usually don't really need the negative gcd so in most books you will see a definition according to which the gcd must be positive.
– Mark
Oct 2 '18 at 12:37
Thank you :) I'm just having a hard time accepting that a particular concept has 2 definition, both of them are slightly different from one another. This just doesn't seem math-likely though...
– William
Oct 2 '18 at 14:55
Well, note that all the definitions of the positive gcd are equivalent, there are no contradictions here. It's just that question-do you want to call the negative number a gcd or not? It's all just about names. I prefer the gcd to be positive and unique, it's much easier to think this way.
– Mark
Oct 2 '18 at 16:40
add a comment |
But then again, if you agree that gcd can be positive as well as negative, obviously, positive > negative. So the positive divisor should be considered as it's greater. How can gcd be negative in the first place, as it asks for the "greatest" factor.
– William
Oct 2 '18 at 12:31
1
So again, it's up to how you define. The definition I gave you obviously requires the gcd to be positive and hence unique. But in other places you might find a definition that doesn't require that and allows the gcd to be negative as well. (your definition for example). So all the question is do you call the negative number a gcd. Some people do, some do not. The fact is we usually don't really need the negative gcd so in most books you will see a definition according to which the gcd must be positive.
– Mark
Oct 2 '18 at 12:37
Thank you :) I'm just having a hard time accepting that a particular concept has 2 definition, both of them are slightly different from one another. This just doesn't seem math-likely though...
– William
Oct 2 '18 at 14:55
Well, note that all the definitions of the positive gcd are equivalent, there are no contradictions here. It's just that question-do you want to call the negative number a gcd or not? It's all just about names. I prefer the gcd to be positive and unique, it's much easier to think this way.
– Mark
Oct 2 '18 at 16:40
But then again, if you agree that gcd can be positive as well as negative, obviously, positive > negative. So the positive divisor should be considered as it's greater. How can gcd be negative in the first place, as it asks for the "greatest" factor.
– William
Oct 2 '18 at 12:31
But then again, if you agree that gcd can be positive as well as negative, obviously, positive > negative. So the positive divisor should be considered as it's greater. How can gcd be negative in the first place, as it asks for the "greatest" factor.
– William
Oct 2 '18 at 12:31
1
1
So again, it's up to how you define. The definition I gave you obviously requires the gcd to be positive and hence unique. But in other places you might find a definition that doesn't require that and allows the gcd to be negative as well. (your definition for example). So all the question is do you call the negative number a gcd. Some people do, some do not. The fact is we usually don't really need the negative gcd so in most books you will see a definition according to which the gcd must be positive.
– Mark
Oct 2 '18 at 12:37
So again, it's up to how you define. The definition I gave you obviously requires the gcd to be positive and hence unique. But in other places you might find a definition that doesn't require that and allows the gcd to be negative as well. (your definition for example). So all the question is do you call the negative number a gcd. Some people do, some do not. The fact is we usually don't really need the negative gcd so in most books you will see a definition according to which the gcd must be positive.
– Mark
Oct 2 '18 at 12:37
Thank you :) I'm just having a hard time accepting that a particular concept has 2 definition, both of them are slightly different from one another. This just doesn't seem math-likely though...
– William
Oct 2 '18 at 14:55
Thank you :) I'm just having a hard time accepting that a particular concept has 2 definition, both of them are slightly different from one another. This just doesn't seem math-likely though...
– William
Oct 2 '18 at 14:55
Well, note that all the definitions of the positive gcd are equivalent, there are no contradictions here. It's just that question-do you want to call the negative number a gcd or not? It's all just about names. I prefer the gcd to be positive and unique, it's much easier to think this way.
– Mark
Oct 2 '18 at 16:40
Well, note that all the definitions of the positive gcd are equivalent, there are no contradictions here. It's just that question-do you want to call the negative number a gcd or not? It's all just about names. I prefer the gcd to be positive and unique, it's much easier to think this way.
– Mark
Oct 2 '18 at 16:40
add a comment |
The text is using the universal definition of a gcd, namely
$$ cmid a,b iff cmid gcd(a,b)$$
Direction $(Leftarrow)$ implies that gcd is a common divisor of $a,b,$ (by choosing $ c = gcd(a,b))$ and the reverse direction $(Rightarrow)$ implies that the gcd is "greatest" w.r.t. divisibility order, i.e. divisible by all other common divisors $c$ of $a,b$.
Scaling the gcd by a unit (invertible) $u$ (e.g. $-1)$ preserves the above definition, so generally gcds are unique only up to unit multiples, i.e. up to associates. In some rings with simple unit group structure we can choose canonical representatives of associate classes, which allows is to choose normal-forms for gcds, e.g. in $,Bbb Z,$ (with units $pm 1)$ we normalize gcds $ge 0,,$ and in a polynomial ring $,K[x],$ over a field (units = constants $0neq cin K) $ we normalize polynomial gcds to be monic (lead coeff $,c_n = 1),,$ by scaling the polynomial by $,c_n^{-1},$ if need be (so a constant gcd $,c_0neq 0$ normalizes to $1).,$ Hence in both cases we can say that two elements are coprime if their gcd $= 1$ (vs. a unit). Such normalizations are sometimes called unit normal representatives in the literature.
answered Oct 2 '18 at 16:00
Bill Dubuque
209k29190629
add a comment |
The text is using the universal definition of a gcd, namely
$$ cmid a,b iff cmid gcd(a,b)$$
Direction $(Leftarrow)$ implies that gcd is a common divisor of $a,b,$ (by choosing $ c = gcd(a,b))$ and the reverse direction $(Rightarrow)$ implies that the gcd is "greatest" w.r.t. divisibility order, i.e. divisible by all other common divisors $c$ of $a,b$.
Scaling the gcd by a unit (invertible) $u$ (e.g. $-1)$ preserves the above definition, so generally gcds are unique only up to unit multiples, i.e. up to associates. In some rings with simple unit group structure we can choose canonical representatives of associate classes, which allows is to choose normal-forms for gcds, e.g. in $,Bbb Z,$ (with units $pm 1)$ we normalize gcds $ge 0,,$ and in a polynomial ring $,K[x],$ over a field (units = constants $0neq cin K) $ we normalize polynomial gcds to be monic (lead coeff $,c_n = 1),,$ by scaling the polynomial by $,c_n^{-1},$ if need be (so a constant gcd $,c_0neq 0$ normalizes to $1).,$ Hence in both cases we can say that two elements are coprime if their gcd $= 1$ (vs. a unit). Such normalizations are sometimes called unit normal representatives in the literature.
answered Oct 2 '18 at 16:00
Bill Dubuque
209k29190629
add a comment |
The text is using the universal definition of a gcd, namely
$$ cmid a,b iff cmid gcd(a,b)$$
Direction $(Leftarrow)$ implies that gcd is a common divisor of $a,b,$ (by choosing $ c = gcd(a,b))$ and the reverse direction $(Rightarrow)$ implies that the gcd is "greatest" w.r.t. divisibility order, i.e. divisible by all other common divisors $c$ of $a,b$.
Scaling the gcd by a unit (invertible) $u$ (e.g. $-1)$ preserves the above definition, so generally gcds are unique only up to unit multiples, i.e. up to associates. In some rings with simple unit group structure we can choose canonical representatives of associate classes, which allows is to choose normal-forms for gcds, e.g. in $,Bbb Z,$ (with units $pm 1)$ we normalize gcds $ge 0,,$ and in a polynomial ring $,K[x],$ over a field (units = constants $0neq cin K) $ we normalize polynomial gcds to be monic (lead coeff $,c_n = 1),,$ by scaling the polynomial by $,c_n^{-1},$ if need be (so a constant gcd $,c_0neq 0$ normalizes to $1).,$ Hence in both cases we can say that two elements are coprime if their gcd $= 1$ (vs. a unit). Such normalizations are sometimes called unit normal representatives in the literature.
answered Oct 2 '18 at 16:00
Bill Dubuque
209k29190629
The text is using the universal definition of a gcd, namely
$$ cmid a,b iff cmid gcd(a,b)$$
Direction $(Leftarrow)$ implies that gcd is a common divisor of $a,b,$ (by choosing $ c = gcd(a,b))$ and the reverse direction $(Rightarrow)$ implies that the gcd is "greatest" w.r.t. divisibility order, i.e. divisible by all other common divisors $c$ of $a,b$.
Scaling the gcd by a unit (invertible) $u$ (e.g. $-1)$ preserves the above definition, so generally gcds are unique only up to unit multiples, i.e. up to associates. In some rings with simple unit group structure we can choose canonical representatives of associate classes, which allows is to choose normal-forms for gcds, e.g. in $,Bbb Z,$ (with units $pm 1)$ we normalize gcds $ge 0,,$ and in a polynomial ring $,K[x],$ over a field (units = constants $0neq cin K) $ we normalize polynomial gcds to be monic (lead coeff $,c_n = 1),,$ by scaling the polynomial by $,c_n^{-1},$ if need be (so a constant gcd $,c_0neq 0$ normalizes to $1).,$ Hence in both cases we can say that two elements are coprime if their gcd $= 1$ (vs. a unit). Such normalizations are sometimes called unit normal representatives in the literature.
answered Oct 2 '18 at 16:00
Bill Dubuque
209k29190629
edited Oct 2 '18 at 16:26
answered Oct 2 '18 at 16:00
Bill Dubuque
209k29190629
answered Oct 2 '18 at 16:00
Bill Dubuque
209k29190629
answered Oct 2 '18 at 16:00
Bill Dubuque
209k29190629
209k29190629
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2
Where is than definition from? From wikipedia: "In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers"
– Roberto Rastapopoulos
Oct 2 '18 at 10:15
I expect your definition required $din mathbb N$,
– lulu
Oct 2 '18 at 10:18
3
Indeed, the greatest common divisor of two elements is only uniquely determined up to an invertible element (i.e. up to a sign in the integers). For the integers we usually choose the positive one, since this is a "nice" way to single out one of them.
– Tobias Kildetoft
Oct 2 '18 at 10:20
@RobertoRastapopoulos you can find it many algebra textbooks, but here is an example in the form of question. math.stackexchange.com/q/85565/552998
– William
Oct 2 '18 at 10:24
2
In some definitions it's a gcd (because there can be more than one)
– John Cataldo
Oct 2 '18 at 10:42