Group automorphisms of the non-zero elements of a field
Clash Royale CLAN TAG#URR8PPP
up vote
4
down vote
favorite
I would like to know what is $Hom_Groups(K^*,K^*)$, at least in the case $K$ is a complete non-archimedean (valued) field. Is this $mathbbZ$?
field-theory abelian-groups
edited Aug 19 at 8:59
Daniel Fischer♦
172k16155276
asked Jan 24 '13 at 15:00
iago
416213
add a comment |Â
up vote
4
down vote
favorite
I would like to know what is $Hom_Groups(K^*,K^*)$, at least in the case $K$ is a complete non-archimedean (valued) field. Is this $mathbbZ$?
field-theory abelian-groups
edited Aug 19 at 8:59
Daniel Fischer♦
172k16155276
asked Jan 24 '13 at 15:00
iago
416213
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I would like to know what is $Hom_Groups(K^*,K^*)$, at least in the case $K$ is a complete non-archimedean (valued) field. Is this $mathbbZ$?
field-theory abelian-groups
edited Aug 19 at 8:59
Daniel Fischer♦
172k16155276
asked Jan 24 '13 at 15:00
iago
416213
I would like to know what is $Hom_Groups(K^*,K^*)$, at least in the case $K$ is a complete non-archimedean (valued) field. Is this $mathbbZ$?
field-theory abelian-groups
edited Aug 19 at 8:59
Daniel Fischer♦
172k16155276
asked Jan 24 '13 at 15:00
iago
416213
edited Aug 19 at 8:59
Daniel Fischer♦
172k16155276
edited Aug 19 at 8:59
Daniel Fischer♦
172k16155276
edited Aug 19 at 8:59
Daniel Fischer♦
172k16155276
172k16155276
asked Jan 24 '13 at 15:00
iago
416213
asked Jan 24 '13 at 15:00
iago
416213
asked Jan 24 '13 at 15:00
iago
416213
416213
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
Much bigger than $mathbb Z$, at least if (as is standard) you mean by “$K^*$†the set of nonzero elements of $K$. (Your title suggests something different.)
I will say something about the automorphisms of a local field of mixed characteristic, i.e. a finite extension of $mathbb Q_p$. The question is certainly much more difficult if you accept discontinuous automorphisms, and I will not consider such. What you need to do is decompose $K^*$ as direct sum of simpler groups, and you find that
$$
K^*cong mathbb Zoplus Woplus(mathbb Z_p)^n>,
$$
where $W$ is the (finite) group of roots of unity of $K$, and $n=[Kcolonmathbb Q_p]$. This decomposition is not unique. Now to see all automorphisms of this group you need to look not only at the automorphisms of the summands but also at the homomorphisms from one summand to another.
answered Jan 24 '13 at 15:26
Lubin
41.3k34184
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Much bigger than $mathbb Z$, at least if (as is standard) you mean by “$K^*$†the set of nonzero elements of $K$. (Your title suggests something different.)
I will say something about the automorphisms of a local field of mixed characteristic, i.e. a finite extension of $mathbb Q_p$. The question is certainly much more difficult if you accept discontinuous automorphisms, and I will not consider such. What you need to do is decompose $K^*$ as direct sum of simpler groups, and you find that
$$
K^*cong mathbb Zoplus Woplus(mathbb Z_p)^n>,
$$
where $W$ is the (finite) group of roots of unity of $K$, and $n=[Kcolonmathbb Q_p]$. This decomposition is not unique. Now to see all automorphisms of this group you need to look not only at the automorphisms of the summands but also at the homomorphisms from one summand to another.
answered Jan 24 '13 at 15:26
Lubin
41.3k34184
add a comment |Â
up vote
3
down vote
accepted
Much bigger than $mathbb Z$, at least if (as is standard) you mean by “$K^*$†the set of nonzero elements of $K$. (Your title suggests something different.)
I will say something about the automorphisms of a local field of mixed characteristic, i.e. a finite extension of $mathbb Q_p$. The question is certainly much more difficult if you accept discontinuous automorphisms, and I will not consider such. What you need to do is decompose $K^*$ as direct sum of simpler groups, and you find that
$$
K^*cong mathbb Zoplus Woplus(mathbb Z_p)^n>,
$$
where $W$ is the (finite) group of roots of unity of $K$, and $n=[Kcolonmathbb Q_p]$. This decomposition is not unique. Now to see all automorphisms of this group you need to look not only at the automorphisms of the summands but also at the homomorphisms from one summand to another.
answered Jan 24 '13 at 15:26
Lubin
41.3k34184
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Much bigger than $mathbb Z$, at least if (as is standard) you mean by “$K^*$†the set of nonzero elements of $K$. (Your title suggests something different.)
I will say something about the automorphisms of a local field of mixed characteristic, i.e. a finite extension of $mathbb Q_p$. The question is certainly much more difficult if you accept discontinuous automorphisms, and I will not consider such. What you need to do is decompose $K^*$ as direct sum of simpler groups, and you find that
$$
K^*cong mathbb Zoplus Woplus(mathbb Z_p)^n>,
$$
where $W$ is the (finite) group of roots of unity of $K$, and $n=[Kcolonmathbb Q_p]$. This decomposition is not unique. Now to see all automorphisms of this group you need to look not only at the automorphisms of the summands but also at the homomorphisms from one summand to another.
answered Jan 24 '13 at 15:26
Lubin
41.3k34184
Much bigger than $mathbb Z$, at least if (as is standard) you mean by “$K^*$†the set of nonzero elements of $K$. (Your title suggests something different.)
I will say something about the automorphisms of a local field of mixed characteristic, i.e. a finite extension of $mathbb Q_p$. The question is certainly much more difficult if you accept discontinuous automorphisms, and I will not consider such. What you need to do is decompose $K^*$ as direct sum of simpler groups, and you find that
$$
K^*cong mathbb Zoplus Woplus(mathbb Z_p)^n>,
$$
where $W$ is the (finite) group of roots of unity of $K$, and $n=[Kcolonmathbb Q_p]$. This decomposition is not unique. Now to see all automorphisms of this group you need to look not only at the automorphisms of the summands but also at the homomorphisms from one summand to another.
answered Jan 24 '13 at 15:26
Lubin
41.3k34184
answered Jan 24 '13 at 15:26
Lubin
41.3k34184
answered Jan 24 '13 at 15:26
Lubin
41.3k34184
answered Jan 24 '13 at 15:26
Lubin
41.3k34184
41.3k34184
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f285882%2fgroup-automorphisms-of-the-non-zero-elements-of-a-field%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
