Finding image of a map from eigen values in $7$-adic number field

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I have been studying algebraic number theory from "$p$-adic Numbers : An Introduction" by Gouvea, along with that some other articles also.

One question has come to my mind, I am trying to express it neatly in step-by-step.

Let $mathbb Q_7$ to be the field of $7$-adic rationals and $mathbb Z_7$ to be its ring of integers.

It is also know that all primitive $6$-th roots of unity is in $mathbb Q_7$ [See Section 4.3 of the above mentioned book]. Let us take $omega$ to be one primitive $6$-th root of unity.

Let $theta$ be a primitive $7$-th root of unity. Consider the group ($mathfrak p^2, +)$ where $mathfrak p^2 = (theta-1)^2 mathbb Z_7[theta]$ is an ideal of $mathbb Z_7[theta]$.

Also suppose $sigma_3$ denotes the ring automorphism of $mathbb Z_7[theta]$ defined by $thetamapsto theta^3$ which is $mathbb Q_7$-linear and it has order $6$. Hence it is diagonisable with eigen values $omega^imid 0le ile 5$.

This allows us to define a $mathbb Q_7$-linear map $tau : mathfrak p^2 to mathfrak p^2$ by $xmapsto sigma_3(x)+sigma_3^5(x)-x$ having the eigen values $omega^i+omega^5i-1mid 0le ile 5 =1,-2,-3,0$ with the algebraic multiplicities $1,2,1,2$ resctively.

Now my question is

Is there any method so that we can describe the image of $tau$, more specifically can we find the generators of the abelian group ($mathfrak p^2/tau(mathfrak p^2)$, +) ?

Sorry for not showing much effort from my side. I kind of understand the existence of eigen value zero might cause problem but I am not sure. Any help will be greatly appreciated.

Thanks in advance.

linear-algebra eigenvalues-eigenvectors linear-transformations algebraic-number-theory p-adic-number-theory

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edited Jul 26 at 6:17

asked Jul 26 at 6:10

usermath

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I have been studying algebraic number theory from "$p$-adic Numbers : An Introduction" by Gouvea, along with that some other articles also.

One question has come to my mind, I am trying to express it neatly in step-by-step.

Let $mathbb Q_7$ to be the field of $7$-adic rationals and $mathbb Z_7$ to be its ring of integers.

It is also know that all primitive $6$-th roots of unity is in $mathbb Q_7$ [See Section 4.3 of the above mentioned book]. Let us take $omega$ to be one primitive $6$-th root of unity.

Let $theta$ be a primitive $7$-th root of unity. Consider the group ($mathfrak p^2, +)$ where $mathfrak p^2 = (theta-1)^2 mathbb Z_7[theta]$ is an ideal of $mathbb Z_7[theta]$.

Also suppose $sigma_3$ denotes the ring automorphism of $mathbb Z_7[theta]$ defined by $thetamapsto theta^3$ which is $mathbb Q_7$-linear and it has order $6$. Hence it is diagonisable with eigen values $omega^imid 0le ile 5$.

This allows us to define a $mathbb Q_7$-linear map $tau : mathfrak p^2 to mathfrak p^2$ by $xmapsto sigma_3(x)+sigma_3^5(x)-x$ having the eigen values $omega^i+omega^5i-1mid 0le ile 5 =1,-2,-3,0$ with the algebraic multiplicities $1,2,1,2$ resctively.

Now my question is

Is there any method so that we can describe the image of $tau$, more specifically can we find the generators of the abelian group ($mathfrak p^2/tau(mathfrak p^2)$, +) ?

Sorry for not showing much effort from my side. I kind of understand the existence of eigen value zero might cause problem but I am not sure. Any help will be greatly appreciated.

Thanks in advance.

linear-algebra eigenvalues-eigenvectors linear-transformations algebraic-number-theory p-adic-number-theory

share|cite|improve this question

edited Jul 26 at 6:17

asked Jul 26 at 6:10

usermath

2,4391227

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

I have been studying algebraic number theory from "$p$-adic Numbers : An Introduction" by Gouvea, along with that some other articles also.

One question has come to my mind, I am trying to express it neatly in step-by-step.

Let $mathbb Q_7$ to be the field of $7$-adic rationals and $mathbb Z_7$ to be its ring of integers.

It is also know that all primitive $6$-th roots of unity is in $mathbb Q_7$ [See Section 4.3 of the above mentioned book]. Let us take $omega$ to be one primitive $6$-th root of unity.

Let $theta$ be a primitive $7$-th root of unity. Consider the group ($mathfrak p^2, +)$ where $mathfrak p^2 = (theta-1)^2 mathbb Z_7[theta]$ is an ideal of $mathbb Z_7[theta]$.

Also suppose $sigma_3$ denotes the ring automorphism of $mathbb Z_7[theta]$ defined by $thetamapsto theta^3$ which is $mathbb Q_7$-linear and it has order $6$. Hence it is diagonisable with eigen values $omega^imid 0le ile 5$.

This allows us to define a $mathbb Q_7$-linear map $tau : mathfrak p^2 to mathfrak p^2$ by $xmapsto sigma_3(x)+sigma_3^5(x)-x$ having the eigen values $omega^i+omega^5i-1mid 0le ile 5 =1,-2,-3,0$ with the algebraic multiplicities $1,2,1,2$ resctively.

Now my question is

Is there any method so that we can describe the image of $tau$, more specifically can we find the generators of the abelian group ($mathfrak p^2/tau(mathfrak p^2)$, +) ?

Sorry for not showing much effort from my side. I kind of understand the existence of eigen value zero might cause problem but I am not sure. Any help will be greatly appreciated.

Thanks in advance.

linear-algebra eigenvalues-eigenvectors linear-transformations algebraic-number-theory p-adic-number-theory

share|cite|improve this question

edited Jul 26 at 6:17

asked Jul 26 at 6:10

usermath

2,4391227

I have been studying algebraic number theory from "$p$-adic Numbers : An Introduction" by Gouvea, along with that some other articles also.

One question has come to my mind, I am trying to express it neatly in step-by-step.

Let $mathbb Q_7$ to be the field of $7$-adic rationals and $mathbb Z_7$ to be its ring of integers.

It is also know that all primitive $6$-th roots of unity is in $mathbb Q_7$ [See Section 4.3 of the above mentioned book]. Let us take $omega$ to be one primitive $6$-th root of unity.

Let $theta$ be a primitive $7$-th root of unity. Consider the group ($mathfrak p^2, +)$ where $mathfrak p^2 = (theta-1)^2 mathbb Z_7[theta]$ is an ideal of $mathbb Z_7[theta]$.

Also suppose $sigma_3$ denotes the ring automorphism of $mathbb Z_7[theta]$ defined by $thetamapsto theta^3$ which is $mathbb Q_7$-linear and it has order $6$. Hence it is diagonisable with eigen values $omega^imid 0le ile 5$.

This allows us to define a $mathbb Q_7$-linear map $tau : mathfrak p^2 to mathfrak p^2$ by $xmapsto sigma_3(x)+sigma_3^5(x)-x$ having the eigen values $omega^i+omega^5i-1mid 0le ile 5 =1,-2,-3,0$ with the algebraic multiplicities $1,2,1,2$ resctively.

Now my question is

Is there any method so that we can describe the image of $tau$, more specifically can we find the generators of the abelian group ($mathfrak p^2/tau(mathfrak p^2)$, +) ?

Sorry for not showing much effort from my side. I kind of understand the existence of eigen value zero might cause problem but I am not sure. Any help will be greatly appreciated.

Thanks in advance.

linear-algebra eigenvalues-eigenvectors linear-transformations algebraic-number-theory p-adic-number-theory

share|cite|improve this question

edited Jul 26 at 6:17

asked Jul 26 at 6:10

usermath

2,4391227

share|cite|improve this question

share|cite|improve this question

edited Jul 26 at 6:17

edited Jul 26 at 6:17

edited Jul 26 at 6:17

asked Jul 26 at 6:10

usermath

2,4391227

asked Jul 26 at 6:10

usermath

2,4391227

asked Jul 26 at 6:10

usermath

2,4391227

2,4391227

add a comment | 

add a comment | 

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