Is the set of the irreducible elements of a field empty? [closed]
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The set of the irreducible elements of a field is empty.
I'm confused with this affirmatiom.
Let's take for example ℚ. Every elements is unit, so every element is a product of unit elements. Every element is irreducible then ?
calculus abstract-algebra
asked Jan 13 at 13:43
Alexia Alexia
31
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closed as unclear what you're asking by Namaste, Dietrich Burde, José Carlos Santos, Xander Henderson, mrtaurho Jan 13 at 21:35
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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 |
$begingroup$
The set of the irreducible elements of a field is empty.
I'm confused with this affirmatiom.
Let's take for example ℚ. Every elements is unit, so every element is a product of unit elements. Every element is irreducible then ?
calculus abstract-algebra
asked Jan 13 at 13:43
Alexia Alexia
31
$endgroup$
closed as unclear what you're asking by Namaste, Dietrich Burde, José Carlos Santos, Xander Henderson, mrtaurho Jan 13 at 21:35
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
2
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I've only seen "reducible" and "irreducible" applied to non-units. I don't think the concept makes sense for units.
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– lulu
Jan 13 at 13:46
1
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What is your definition of irreducible? All references I know define irreducible to non-zeros non-unity.
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– Lucas Corrêa
Jan 13 at 13:53
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An element e in A is irreducible if 1) e is not a unit in A (e is not invertible in A), 2) if e = ab, then a or b is a unit.
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– Alexia
Jan 13 at 13:55
add a comment |
$begingroup$
The set of the irreducible elements of a field is empty.
I'm confused with this affirmatiom.
Let's take for example ℚ. Every elements is unit, so every element is a product of unit elements. Every element is irreducible then ?
calculus abstract-algebra
asked Jan 13 at 13:43
Alexia Alexia
31
$endgroup$
The set of the irreducible elements of a field is empty.
I'm confused with this affirmatiom.
Let's take for example ℚ. Every elements is unit, so every element is a product of unit elements. Every element is irreducible then ?
calculus abstract-algebra
calculus abstract-algebra
asked Jan 13 at 13:43
Alexia Alexia
31
asked Jan 13 at 13:43
Alexia Alexia
31
asked Jan 13 at 13:43
Alexia Alexia
31
asked Jan 13 at 13:43
Alexia Alexia
31
asked Jan 13 at 13:43
Alexia Alexia
31
31
closed as unclear what you're asking by Namaste, Dietrich Burde, José Carlos Santos, Xander Henderson, mrtaurho Jan 13 at 21:35
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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 unclear what you're asking by Namaste, Dietrich Burde, José Carlos Santos, Xander Henderson, mrtaurho Jan 13 at 21:35
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
2
$begingroup$
I've only seen "reducible" and "irreducible" applied to non-units. I don't think the concept makes sense for units.
$endgroup$
– lulu
Jan 13 at 13:46
1
$begingroup$
What is your definition of irreducible? All references I know define irreducible to non-zeros non-unity.
$endgroup$
– Lucas Corrêa
Jan 13 at 13:53
$begingroup$
An element e in A is irreducible if 1) e is not a unit in A (e is not invertible in A), 2) if e = ab, then a or b is a unit.
$endgroup$
– Alexia
Jan 13 at 13:55
add a comment |
2
$begingroup$
I've only seen "reducible" and "irreducible" applied to non-units. I don't think the concept makes sense for units.
$endgroup$
– lulu
Jan 13 at 13:46
1
$begingroup$
What is your definition of irreducible? All references I know define irreducible to non-zeros non-unity.
$endgroup$
– Lucas Corrêa
Jan 13 at 13:53
$begingroup$
An element e in A is irreducible if 1) e is not a unit in A (e is not invertible in A), 2) if e = ab, then a or b is a unit.
$endgroup$
– Alexia
Jan 13 at 13:55
2
2
$begingroup$
I've only seen "reducible" and "irreducible" applied to non-units. I don't think the concept makes sense for units.
$endgroup$
– lulu
Jan 13 at 13:46
$begingroup$
I've only seen "reducible" and "irreducible" applied to non-units. I don't think the concept makes sense for units.
$endgroup$
– lulu
Jan 13 at 13:46
1
1
$begingroup$
What is your definition of irreducible? All references I know define irreducible to non-zeros non-unity.
$endgroup$
– Lucas Corrêa
Jan 13 at 13:53
$begingroup$
What is your definition of irreducible? All references I know define irreducible to non-zeros non-unity.
$endgroup$
– Lucas Corrêa
Jan 13 at 13:53
$begingroup$
An element e in A is irreducible if 1) e is not a unit in A (e is not invertible in A), 2) if e = ab, then a or b is a unit.
$endgroup$
– Alexia
Jan 13 at 13:55
$begingroup$
An element e in A is irreducible if 1) e is not a unit in A (e is not invertible in A), 2) if e = ab, then a or b is a unit.
$endgroup$
– Alexia
Jan 13 at 13:55
add a comment |
2 Answers
2
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An irreducible element is a nonzero nonunit element $x$ such that if $x=ab$ for some nonunit $a$, then $b$ is a unit. In a field there are no nonzero, nonunit elements, so there are no irreducible elements.
answered Jan 13 at 13:54
Matt SamuelMatt Samuel
38.8k63769
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Oh, my bad. I forgot the first condition of a nonunit element x. Thank you.
$endgroup$
– Alexia
Jan 13 at 13:58
add a comment |
$begingroup$
By definition, a non-zero non-unit element in an integral domain is called irreducible if it is not a product of two non-units.
But in a field, all non-zero elements are units. So there are no irreducible elements in a field.
answered Jan 13 at 13:54
WuestenfuxWuestenfux
5,1371513
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add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
An irreducible element is a nonzero nonunit element $x$ such that if $x=ab$ for some nonunit $a$, then $b$ is a unit. In a field there are no nonzero, nonunit elements, so there are no irreducible elements.
answered Jan 13 at 13:54
Matt SamuelMatt Samuel
38.8k63769
$endgroup$
$begingroup$
Oh, my bad. I forgot the first condition of a nonunit element x. Thank you.
$endgroup$
– Alexia
Jan 13 at 13:58
add a comment |
$begingroup$
An irreducible element is a nonzero nonunit element $x$ such that if $x=ab$ for some nonunit $a$, then $b$ is a unit. In a field there are no nonzero, nonunit elements, so there are no irreducible elements.
answered Jan 13 at 13:54
Matt SamuelMatt Samuel
38.8k63769
$endgroup$
$begingroup$
Oh, my bad. I forgot the first condition of a nonunit element x. Thank you.
$endgroup$
– Alexia
Jan 13 at 13:58
add a comment |
$begingroup$
An irreducible element is a nonzero nonunit element $x$ such that if $x=ab$ for some nonunit $a$, then $b$ is a unit. In a field there are no nonzero, nonunit elements, so there are no irreducible elements.
answered Jan 13 at 13:54
Matt SamuelMatt Samuel
38.8k63769
$endgroup$
An irreducible element is a nonzero nonunit element $x$ such that if $x=ab$ for some nonunit $a$, then $b$ is a unit. In a field there are no nonzero, nonunit elements, so there are no irreducible elements.
answered Jan 13 at 13:54
Matt SamuelMatt Samuel
38.8k63769
answered Jan 13 at 13:54
Matt SamuelMatt Samuel
38.8k63769
answered Jan 13 at 13:54
Matt SamuelMatt Samuel
38.8k63769
answered Jan 13 at 13:54
Matt SamuelMatt Samuel
38.8k63769
38.8k63769
$begingroup$
Oh, my bad. I forgot the first condition of a nonunit element x. Thank you.
$endgroup$
– Alexia
Jan 13 at 13:58
add a comment |
$begingroup$
Oh, my bad. I forgot the first condition of a nonunit element x. Thank you.
$endgroup$
– Alexia
Jan 13 at 13:58
$begingroup$
Oh, my bad. I forgot the first condition of a nonunit element x. Thank you.
$endgroup$
– Alexia
Jan 13 at 13:58
$begingroup$
Oh, my bad. I forgot the first condition of a nonunit element x. Thank you.
$endgroup$
– Alexia
Jan 13 at 13:58
add a comment |
$begingroup$
By definition, a non-zero non-unit element in an integral domain is called irreducible if it is not a product of two non-units.
But in a field, all non-zero elements are units. So there are no irreducible elements in a field.
answered Jan 13 at 13:54
WuestenfuxWuestenfux
5,1371513
$endgroup$
add a comment |
$begingroup$
By definition, a non-zero non-unit element in an integral domain is called irreducible if it is not a product of two non-units.
But in a field, all non-zero elements are units. So there are no irreducible elements in a field.
answered Jan 13 at 13:54
WuestenfuxWuestenfux
5,1371513
$endgroup$
add a comment |
$begingroup$
By definition, a non-zero non-unit element in an integral domain is called irreducible if it is not a product of two non-units.
But in a field, all non-zero elements are units. So there are no irreducible elements in a field.
answered Jan 13 at 13:54
WuestenfuxWuestenfux
5,1371513
$endgroup$
By definition, a non-zero non-unit element in an integral domain is called irreducible if it is not a product of two non-units.
But in a field, all non-zero elements are units. So there are no irreducible elements in a field.
answered Jan 13 at 13:54
WuestenfuxWuestenfux
5,1371513
answered Jan 13 at 13:54
WuestenfuxWuestenfux
5,1371513
answered Jan 13 at 13:54
WuestenfuxWuestenfux
5,1371513
answered Jan 13 at 13:54
WuestenfuxWuestenfux
5,1371513
5,1371513
add a comment |
add a comment |
2
$begingroup$
I've only seen "reducible" and "irreducible" applied to non-units. I don't think the concept makes sense for units.
$endgroup$
– lulu
Jan 13 at 13:46
1
$begingroup$
What is your definition of irreducible? All references I know define irreducible to non-zeros non-unity.
$endgroup$
– Lucas Corrêa
Jan 13 at 13:53
$begingroup$
An element e in A is irreducible if 1) e is not a unit in A (e is not invertible in A), 2) if e = ab, then a or b is a unit.
$endgroup$
– Alexia
Jan 13 at 13:55