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Possible sizes of discriminant of some extension
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I know that for a quadratic extension of $mathbbQ$, say $K=mathbbQ(sqrtd)$ where $d$ is square-free, then the discriminant, $Delta_K$, can take two forms:
$$Delta= begincasesd &textif dequiv 1mod4 \ 4d & textif d equiv 2,3 mod4endcases.$$
Are there easy ways to extend this to higher degree extensions? For example, suppose $F$ is a degree two extension of $K = mathbbQ(sqrtd)$, and so a degree 4 extension of $mathbbQ$.
Can $Delta_F$ be arbitrarily large, or does it have some form like
$Delta_K$ does?
I don't see a way to proceed just using the definition of $Delta$ as the determinant of a matrix of embeddings evaluated at the basis elements of $mathcalO_F$ (which is the only one I know at present).
algebraic-number-theory
edited Jul 18 at 22:21
Chappers
55k74190
asked Jul 18 at 21:30
lkr
1068
add a comment |Â
up vote
1
down vote
favorite
I know that for a quadratic extension of $mathbbQ$, say $K=mathbbQ(sqrtd)$ where $d$ is square-free, then the discriminant, $Delta_K$, can take two forms:
$$Delta= begincasesd &textif dequiv 1mod4 \ 4d & textif d equiv 2,3 mod4endcases.$$
Are there easy ways to extend this to higher degree extensions? For example, suppose $F$ is a degree two extension of $K = mathbbQ(sqrtd)$, and so a degree 4 extension of $mathbbQ$.
Can $Delta_F$ be arbitrarily large, or does it have some form like
$Delta_K$ does?
I don't see a way to proceed just using the definition of $Delta$ as the determinant of a matrix of embeddings evaluated at the basis elements of $mathcalO_F$ (which is the only one I know at present).
algebraic-number-theory
edited Jul 18 at 22:21
Chappers
55k74190
asked Jul 18 at 21:30
lkr
1068
Minkowski's bound gives a lower bound on the discriminant in terms of the degree of the extension en.wikipedia.org/wiki/Minkowski%27s_bound
– Tob Ernack
Jul 18 at 22:19
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I know that for a quadratic extension of $mathbbQ$, say $K=mathbbQ(sqrtd)$ where $d$ is square-free, then the discriminant, $Delta_K$, can take two forms:
$$Delta= begincasesd &textif dequiv 1mod4 \ 4d & textif d equiv 2,3 mod4endcases.$$
Are there easy ways to extend this to higher degree extensions? For example, suppose $F$ is a degree two extension of $K = mathbbQ(sqrtd)$, and so a degree 4 extension of $mathbbQ$.
Can $Delta_F$ be arbitrarily large, or does it have some form like
$Delta_K$ does?
I don't see a way to proceed just using the definition of $Delta$ as the determinant of a matrix of embeddings evaluated at the basis elements of $mathcalO_F$ (which is the only one I know at present).
algebraic-number-theory
edited Jul 18 at 22:21
Chappers
55k74190
asked Jul 18 at 21:30
lkr
1068
I know that for a quadratic extension of $mathbbQ$, say $K=mathbbQ(sqrtd)$ where $d$ is square-free, then the discriminant, $Delta_K$, can take two forms:
$$Delta= begincasesd &textif dequiv 1mod4 \ 4d & textif d equiv 2,3 mod4endcases.$$
Are there easy ways to extend this to higher degree extensions? For example, suppose $F$ is a degree two extension of $K = mathbbQ(sqrtd)$, and so a degree 4 extension of $mathbbQ$.
Can $Delta_F$ be arbitrarily large, or does it have some form like
$Delta_K$ does?
I don't see a way to proceed just using the definition of $Delta$ as the determinant of a matrix of embeddings evaluated at the basis elements of $mathcalO_F$ (which is the only one I know at present).
algebraic-number-theory
edited Jul 18 at 22:21
Chappers
55k74190
asked Jul 18 at 21:30
lkr
1068
edited Jul 18 at 22:21
Chappers
55k74190
edited Jul 18 at 22:21
Chappers
55k74190
edited Jul 18 at 22:21
Chappers
55k74190
55k74190
asked Jul 18 at 21:30
lkr
1068
asked Jul 18 at 21:30
lkr
1068
asked Jul 18 at 21:30
lkr
1068
1068
Minkowski's bound gives a lower bound on the discriminant in terms of the degree of the extension en.wikipedia.org/wiki/Minkowski%27s_bound
– Tob Ernack
Jul 18 at 22:19
add a comment |Â
Minkowski's bound gives a lower bound on the discriminant in terms of the degree of the extension en.wikipedia.org/wiki/Minkowski%27s_bound
– Tob Ernack
Jul 18 at 22:19
Minkowski's bound gives a lower bound on the discriminant in terms of the degree of the extension en.wikipedia.org/wiki/Minkowski%27s_bound
– Tob Ernack
Jul 18 at 22:19
Minkowski's bound gives a lower bound on the discriminant in terms of the degree of the extension en.wikipedia.org/wiki/Minkowski%27s_bound
– Tob Ernack
Jul 18 at 22:19
add a comment |Â
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Minkowski's bound gives a lower bound on the discriminant in terms of the degree of the extension en.wikipedia.org/wiki/Minkowski%27s_bound
– Tob Ernack
Jul 18 at 22:19