Talk:Algebraic number field

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The example section of this article says "The smallest and most basic number field is the field Q of rational numbers." From which one is tempted to conclude that finite fields do not exist. 68.167.188.246 (talk) 15:23, 16 April 2010 (UTC)[reply]

Well, finite fields are not number fields. Jakob.scholbach (talk) 16:33, 16 April 2010 (UTC)[reply]

Different nitpick, but related. The Gaussian field "...form the first nontrivial example of a number field." In what sense, "first?" Certainly not in any ordering of fields, but since the previous example talks about the rationals being the smallest, a reader might be confused. I suspect the author means first, historically, or perhaps simply first in terms of encounters in education, but the former seems like a statement of fact, while the latter will depend entirely on education. (People will probably encounter complex numbers first,, but perhaps not the Gaussian number field.) Thomaso (talk) 16:25, 27 June 2013 (UTC)[reply]

"Moreover, a power basis obtained this way can be turned into an integral basis: if the ..."

Are you sure about this? I thought number fields don't necessarily have power integral basis?

http://en.wikipedia.org/wiki/Monogenic_field

93.172.25.96 (talk) 22:55, 4 May 2010 (UTC)[reply]

Quite right, I'll fix this. Thanks. RobHar (talk) 02:41, 5 May 2010 (UTC)[reply]

The article has both forms Archimedean and archimedean, this should probably be made consistent.Wishcow 15:38, 12 February 2016 (UTC) — Preceding unsigned comment added by Wishcow (talk • contribs)

Perhaps a disambiguation is needed because it may be confused with the field of algebraic numbers which is treated *here* --Little bishop (talk) 12:58, 25 September 2019 (UTC)[reply]

In Algebraic_number_field#Ramification it says now

according to the Lasker–Noether theorem (see above),

but there is no cross-reference, and these names are not mentioned anywhere.

Moreover, I do not see the most important property mentioned: that the ramification indices (as well as local degrees f_j) do not depend on j for normal extensions. --Ilya-zz (talk) 06:04, 6 March 2021 (UTC)[reply]