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There exists a well known concrete bound on the discriminant of a number field by Minkowski.

Are there any concrete (completely explicit) improvements of this bound?

I know of a bound by Odlyzko, but it is only asymptotic, and thus not explicit/concrete enough. In particular, I do not know how to apply this bound to a number field of degree (say) $7$.

That is, I am looking for a bound on the discriminant as a function of the degree $n$ of the number field over $\mathbb{Q}$ (improving Minkowski's bound even if $n$ is not too large). I am most interested in the case of at most two real embeddings.

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    $\begingroup$ Dear @Pablo, I might be misremembering, but I think Odlyzko proved an explicit lower bound for the root discriminant of a number field (i.e., not just an asymptotic result). See Theorem 1.(2) in Lower bounds for discriminants of number fields. II. Tôhoku Math. J., 29(2):209–216, 1977 . This precise result of Odlyzko was used by Takeuchi in his paper Arithmetic Fuchsian groups with signature (1; e). J. Math. Soc. Japan, 35(3):381– 407, 1983. to prove finiteness results for arithmetic groups. $\endgroup$
    – Ariyan Javanpeykar
    Commented Oct 6, 2016 at 12:18
  • $\begingroup$ @AriyanJavanpeykar You are right, but the bounds he gives involve some zeta values, so in particular they depend not only on $n$ (or the signature). This makes it impossible for me to effectively bound the discriminant of a general degree $7$ number field. $\endgroup$
    – Pablo
    Commented Oct 6, 2016 at 12:27
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    $\begingroup$ Have a look at Prop. 2.3 in Takeuchi's paper. He proves $d(k) > a^n \exp(-b)$, where $a = 29.099$ and $b=8.3185$. (Here $n$ is the degree of $k$ and $d(k)$ is the discriminant of the totally real number field $k$.) $\endgroup$
    – Ariyan Javanpeykar
    Commented Oct 6, 2016 at 12:45
  • $\begingroup$ @AriyanJavanpeykar thanks for pointing this out, that's cool! Actually (as you can see in my question) I am mostly interested in the non-totally real case. $\endgroup$
    – Pablo
    Commented Oct 6, 2016 at 12:51
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    $\begingroup$ Sorry. I somehow misread that part of your question. $\endgroup$
    – Ariyan Javanpeykar
    Commented Oct 6, 2016 at 15:10

1 Answer 1

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Theorem 2.4 of the paper Local corrections of discriminant bounds and small degree extensions of quadratic base fields by Brueggeman and Doud gives two lower bounds for the discriminant of a number field of arbitrary signature. The formula is a bit complicated, but can easily be computed with a computer algebra system like SAGE. Note that the formula also makes use of Poitou's refinement of the Odlyzko bounds in that the bound incorporates knowledge about the existence of primes of small norm. (For instance, the bound implies that a degree 10 number field with 2 real embeddings has root discriminant at least 5.834, whereas a degree 10 number field with 2 real embeddings in which 2 splits completely has root discriminant at least 28.951.)

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