Show that (x+y)^3 is not equal to x^3+y^3 for some x and y in a field F
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Problem
Let F be a finite field of characteristic 2 with more than two elements. Show that $(x+y)^3 neq x^3+y^3$ for some $x,y in F$
Doubt
If $2x=0$ for all $x in F$ ,then
$(x+y)^3=x^3+y^3+3xxy+3yyx
=x^3+y^3+xxy+yyx$
Any suggestion or hint what to do after this .
abstract-algebra field-theory
asked Sep 6 at 4:38
blue boy
1,117513
add a comment |Â
up vote
1
down vote
favorite
Problem
Let F be a finite field of characteristic 2 with more than two elements. Show that $(x+y)^3 neq x^3+y^3$ for some $x,y in F$
Doubt
If $2x=0$ for all $x in F$ ,then
$(x+y)^3=x^3+y^3+3xxy+3yyx
=x^3+y^3+xxy+yyx$
Any suggestion or hint what to do after this .
abstract-algebra field-theory
asked Sep 6 at 4:38
blue boy
1,117513
Compare what you have to $x^3+y^3$, as the problem tells you to, and show that they aren't always equal.
– Arthur
Sep 6 at 5:04
@Arthur The problem is about understanding a specific field of characteristic $2$ and a specific example. Your hint is unhelpful.
– David Hill
Sep 6 at 5:09
2
@DavidHill Clearly, the simplification of $(x+y)^3$ cannot go any further, and it's time to start comparing the two expressions. The OP might not know that, as he seems stuck, and thus I point out what I think it's the best next step. Who are you to decide that that is unhelpful?
– Arthur
Sep 6 at 5:28
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Problem
Let F be a finite field of characteristic 2 with more than two elements. Show that $(x+y)^3 neq x^3+y^3$ for some $x,y in F$
Doubt
If $2x=0$ for all $x in F$ ,then
$(x+y)^3=x^3+y^3+3xxy+3yyx
=x^3+y^3+xxy+yyx$
Any suggestion or hint what to do after this .
abstract-algebra field-theory
asked Sep 6 at 4:38
blue boy
1,117513
Problem
Let F be a finite field of characteristic 2 with more than two elements. Show that $(x+y)^3 neq x^3+y^3$ for some $x,y in F$
Doubt
If $2x=0$ for all $x in F$ ,then
$(x+y)^3=x^3+y^3+3xxy+3yyx
=x^3+y^3+xxy+yyx$
Any suggestion or hint what to do after this .
abstract-algebra field-theory
abstract-algebra field-theory
asked Sep 6 at 4:38
blue boy
1,117513
asked Sep 6 at 4:38
blue boy
1,117513
edited Sep 6 at 4:43
asked Sep 6 at 4:38
blue boy
1,117513
asked Sep 6 at 4:38
blue boy
1,117513
asked Sep 6 at 4:38
blue boy
1,117513
1,117513
Compare what you have to $x^3+y^3$, as the problem tells you to, and show that they aren't always equal.
– Arthur
Sep 6 at 5:04
@Arthur The problem is about understanding a specific field of characteristic $2$ and a specific example. Your hint is unhelpful.
– David Hill
Sep 6 at 5:09
2
@DavidHill Clearly, the simplification of $(x+y)^3$ cannot go any further, and it's time to start comparing the two expressions. The OP might not know that, as he seems stuck, and thus I point out what I think it's the best next step. Who are you to decide that that is unhelpful?
– Arthur
Sep 6 at 5:28
add a comment |Â
Compare what you have to $x^3+y^3$, as the problem tells you to, and show that they aren't always equal.
– Arthur
Sep 6 at 5:04
@Arthur The problem is about understanding a specific field of characteristic $2$ and a specific example. Your hint is unhelpful.
– David Hill
Sep 6 at 5:09
2
@DavidHill Clearly, the simplification of $(x+y)^3$ cannot go any further, and it's time to start comparing the two expressions. The OP might not know that, as he seems stuck, and thus I point out what I think it's the best next step. Who are you to decide that that is unhelpful?
– Arthur
Sep 6 at 5:28
Compare what you have to $x^3+y^3$, as the problem tells you to, and show that they aren't always equal.
– Arthur
Sep 6 at 5:04
Compare what you have to $x^3+y^3$, as the problem tells you to, and show that they aren't always equal.
– Arthur
Sep 6 at 5:04
@Arthur The problem is about understanding a specific field of characteristic $2$ and a specific example. Your hint is unhelpful.
– David Hill
Sep 6 at 5:09
@Arthur The problem is about understanding a specific field of characteristic $2$ and a specific example. Your hint is unhelpful.
– David Hill
Sep 6 at 5:09
2
2
@DavidHill Clearly, the simplification of $(x+y)^3$ cannot go any further, and it's time to start comparing the two expressions. The OP might not know that, as he seems stuck, and thus I point out what I think it's the best next step. Who are you to decide that that is unhelpful?
– Arthur
Sep 6 at 5:28
@DavidHill Clearly, the simplification of $(x+y)^3$ cannot go any further, and it's time to start comparing the two expressions. The OP might not know that, as he seems stuck, and thus I point out what I think it's the best next step. Who are you to decide that that is unhelpful?
– Arthur
Sep 6 at 5:28
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
3
down vote
accepted
First, note that
$$
x^3 + y^3 = (x + y)^3 iff 0 = xy(x + y).
$$
So, you need to choose $x$ and $y$ such that both $x$ and $y$ are nonzero, and $x + yneq 0.$ Can you show that you can always choose such elements in a characteristic two field $FnotcongBbb F_2$? Hint: Start with $x = 1.$ What can't $y$ be?
answered Sep 6 at 5:14
Stahl
15.7k43351
Reading field for the first time . My concepts are little shaky on field extension. Will come back at your suggestion after reading a bit more.
– blue boy
Sep 6 at 5:16
1
@blueboy Try thinking about it like this. What goes wrong for a field with 2 elements? (Remember that the only such field is $Bbb F_2 = 0,1$.) Then think about how adding another element $alphaneq 0,1$ can fix the problem...
– Stahl
Sep 6 at 5:18
Any $alpha neq -1$ will do .right?
– blue boy
Sep 6 at 5:33
2
@blueboy Remember that $-1 = 1$ in a field of characteristic $2.$ And $alpha$ better not be zero either!
– Stahl
Sep 6 at 5:34
Ok. Thanks for the help.
– blue boy
Sep 6 at 5:35
add a comment |Â
up vote
0
down vote
Well, you are going to have to consider a field extension. The polynomial $t^2+t+1inmathbbF_2[t]$ is irreducible. What equations can you try in $F=mathbbF_2[t]/(t^2+t+1)$?
answered Sep 6 at 5:08
David Hill
8,2231618
1
It looks like this question is about an arbitrary field of characteristic $2$ which is not $Bbb F_2.$ Examining one specific field extension doesn't seem to be what the problem is after here.
– Stahl
Sep 6 at 5:16
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
First, note that
$$
x^3 + y^3 = (x + y)^3 iff 0 = xy(x + y).
$$
So, you need to choose $x$ and $y$ such that both $x$ and $y$ are nonzero, and $x + yneq 0.$ Can you show that you can always choose such elements in a characteristic two field $FnotcongBbb F_2$? Hint: Start with $x = 1.$ What can't $y$ be?
answered Sep 6 at 5:14
Stahl
15.7k43351
Reading field for the first time . My concepts are little shaky on field extension. Will come back at your suggestion after reading a bit more.
– blue boy
Sep 6 at 5:16
1
@blueboy Try thinking about it like this. What goes wrong for a field with 2 elements? (Remember that the only such field is $Bbb F_2 = 0,1$.) Then think about how adding another element $alphaneq 0,1$ can fix the problem...
– Stahl
Sep 6 at 5:18
Any $alpha neq -1$ will do .right?
– blue boy
Sep 6 at 5:33
2
@blueboy Remember that $-1 = 1$ in a field of characteristic $2.$ And $alpha$ better not be zero either!
– Stahl
Sep 6 at 5:34
Ok. Thanks for the help.
– blue boy
Sep 6 at 5:35
add a comment |Â
up vote
3
down vote
accepted
First, note that
$$
x^3 + y^3 = (x + y)^3 iff 0 = xy(x + y).
$$
So, you need to choose $x$ and $y$ such that both $x$ and $y$ are nonzero, and $x + yneq 0.$ Can you show that you can always choose such elements in a characteristic two field $FnotcongBbb F_2$? Hint: Start with $x = 1.$ What can't $y$ be?
answered Sep 6 at 5:14
Stahl
15.7k43351
Reading field for the first time . My concepts are little shaky on field extension. Will come back at your suggestion after reading a bit more.
– blue boy
Sep 6 at 5:16
1
@blueboy Try thinking about it like this. What goes wrong for a field with 2 elements? (Remember that the only such field is $Bbb F_2 = 0,1$.) Then think about how adding another element $alphaneq 0,1$ can fix the problem...
– Stahl
Sep 6 at 5:18
Any $alpha neq -1$ will do .right?
– blue boy
Sep 6 at 5:33
2
@blueboy Remember that $-1 = 1$ in a field of characteristic $2.$ And $alpha$ better not be zero either!
– Stahl
Sep 6 at 5:34
Ok. Thanks for the help.
– blue boy
Sep 6 at 5:35
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
First, note that
$$
x^3 + y^3 = (x + y)^3 iff 0 = xy(x + y).
$$
So, you need to choose $x$ and $y$ such that both $x$ and $y$ are nonzero, and $x + yneq 0.$ Can you show that you can always choose such elements in a characteristic two field $FnotcongBbb F_2$? Hint: Start with $x = 1.$ What can't $y$ be?
answered Sep 6 at 5:14
Stahl
15.7k43351
First, note that
$$
x^3 + y^3 = (x + y)^3 iff 0 = xy(x + y).
$$
So, you need to choose $x$ and $y$ such that both $x$ and $y$ are nonzero, and $x + yneq 0.$ Can you show that you can always choose such elements in a characteristic two field $FnotcongBbb F_2$? Hint: Start with $x = 1.$ What can't $y$ be?
answered Sep 6 at 5:14
Stahl
15.7k43351
answered Sep 6 at 5:14
Stahl
15.7k43351
answered Sep 6 at 5:14
Stahl
15.7k43351
answered Sep 6 at 5:14
Stahl
15.7k43351
15.7k43351
Reading field for the first time . My concepts are little shaky on field extension. Will come back at your suggestion after reading a bit more.
– blue boy
Sep 6 at 5:16
1
@blueboy Try thinking about it like this. What goes wrong for a field with 2 elements? (Remember that the only such field is $Bbb F_2 = 0,1$.) Then think about how adding another element $alphaneq 0,1$ can fix the problem...
– Stahl
Sep 6 at 5:18
Any $alpha neq -1$ will do .right?
– blue boy
Sep 6 at 5:33
2
@blueboy Remember that $-1 = 1$ in a field of characteristic $2.$ And $alpha$ better not be zero either!
– Stahl
Sep 6 at 5:34
Ok. Thanks for the help.
– blue boy
Sep 6 at 5:35
add a comment |Â
Reading field for the first time . My concepts are little shaky on field extension. Will come back at your suggestion after reading a bit more.
– blue boy
Sep 6 at 5:16
1
@blueboy Try thinking about it like this. What goes wrong for a field with 2 elements? (Remember that the only such field is $Bbb F_2 = 0,1$.) Then think about how adding another element $alphaneq 0,1$ can fix the problem...
– Stahl
Sep 6 at 5:18
Any $alpha neq -1$ will do .right?
– blue boy
Sep 6 at 5:33
2
@blueboy Remember that $-1 = 1$ in a field of characteristic $2.$ And $alpha$ better not be zero either!
– Stahl
Sep 6 at 5:34
Ok. Thanks for the help.
– blue boy
Sep 6 at 5:35
Reading field for the first time . My concepts are little shaky on field extension. Will come back at your suggestion after reading a bit more.
– blue boy
Sep 6 at 5:16
Reading field for the first time . My concepts are little shaky on field extension. Will come back at your suggestion after reading a bit more.
– blue boy
Sep 6 at 5:16
1
1
@blueboy Try thinking about it like this. What goes wrong for a field with 2 elements? (Remember that the only such field is $Bbb F_2 = 0,1$.) Then think about how adding another element $alphaneq 0,1$ can fix the problem...
– Stahl
Sep 6 at 5:18
@blueboy Try thinking about it like this. What goes wrong for a field with 2 elements? (Remember that the only such field is $Bbb F_2 = 0,1$.) Then think about how adding another element $alphaneq 0,1$ can fix the problem...
– Stahl
Sep 6 at 5:18
Any $alpha neq -1$ will do .right?
– blue boy
Sep 6 at 5:33
Any $alpha neq -1$ will do .right?
– blue boy
Sep 6 at 5:33
2
2
@blueboy Remember that $-1 = 1$ in a field of characteristic $2.$ And $alpha$ better not be zero either!
– Stahl
Sep 6 at 5:34
@blueboy Remember that $-1 = 1$ in a field of characteristic $2.$ And $alpha$ better not be zero either!
– Stahl
Sep 6 at 5:34
Ok. Thanks for the help.
– blue boy
Sep 6 at 5:35
Ok. Thanks for the help.
– blue boy
Sep 6 at 5:35
add a comment |Â
up vote
0
down vote
Well, you are going to have to consider a field extension. The polynomial $t^2+t+1inmathbbF_2[t]$ is irreducible. What equations can you try in $F=mathbbF_2[t]/(t^2+t+1)$?
answered Sep 6 at 5:08
David Hill
8,2231618
1
It looks like this question is about an arbitrary field of characteristic $2$ which is not $Bbb F_2.$ Examining one specific field extension doesn't seem to be what the problem is after here.
– Stahl
Sep 6 at 5:16
add a comment |Â
up vote
0
down vote
Well, you are going to have to consider a field extension. The polynomial $t^2+t+1inmathbbF_2[t]$ is irreducible. What equations can you try in $F=mathbbF_2[t]/(t^2+t+1)$?
answered Sep 6 at 5:08
David Hill
8,2231618
1
It looks like this question is about an arbitrary field of characteristic $2$ which is not $Bbb F_2.$ Examining one specific field extension doesn't seem to be what the problem is after here.
– Stahl
Sep 6 at 5:16
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Well, you are going to have to consider a field extension. The polynomial $t^2+t+1inmathbbF_2[t]$ is irreducible. What equations can you try in $F=mathbbF_2[t]/(t^2+t+1)$?
answered Sep 6 at 5:08
David Hill
8,2231618
Well, you are going to have to consider a field extension. The polynomial $t^2+t+1inmathbbF_2[t]$ is irreducible. What equations can you try in $F=mathbbF_2[t]/(t^2+t+1)$?
answered Sep 6 at 5:08
David Hill
8,2231618
answered Sep 6 at 5:08
David Hill
8,2231618
answered Sep 6 at 5:08
David Hill
8,2231618
answered Sep 6 at 5:08
David Hill
8,2231618
8,2231618
1
It looks like this question is about an arbitrary field of characteristic $2$ which is not $Bbb F_2.$ Examining one specific field extension doesn't seem to be what the problem is after here.
– Stahl
Sep 6 at 5:16
add a comment |Â
1
It looks like this question is about an arbitrary field of characteristic $2$ which is not $Bbb F_2.$ Examining one specific field extension doesn't seem to be what the problem is after here.
– Stahl
Sep 6 at 5:16
1
1
It looks like this question is about an arbitrary field of characteristic $2$ which is not $Bbb F_2.$ Examining one specific field extension doesn't seem to be what the problem is after here.
– Stahl
Sep 6 at 5:16
It looks like this question is about an arbitrary field of characteristic $2$ which is not $Bbb F_2.$ Examining one specific field extension doesn't seem to be what the problem is after here.
– Stahl
Sep 6 at 5:16
add a comment |Â
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Compare what you have to $x^3+y^3$, as the problem tells you to, and show that they aren't always equal.
– Arthur
Sep 6 at 5:04
@Arthur The problem is about understanding a specific field of characteristic $2$ and a specific example. Your hint is unhelpful.
– David Hill
Sep 6 at 5:09
2
@DavidHill Clearly, the simplification of $(x+y)^3$ cannot go any further, and it's time to start comparing the two expressions. The OP might not know that, as he seems stuck, and thus I point out what I think it's the best next step. Who are you to decide that that is unhelpful?
– Arthur
Sep 6 at 5:28