純粋・応用数学（含むガロア理論）3

[過去ﾛｸﾞ] 純粋・応用数学（含むガロア理論）3 (1002ﾚｽ)
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449: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/20(木) 00:24:25 ID:gmO23IhH

>>448

つづき

諸概念
体 K の乗法群の任意の有限部分群は巡回群である。

体の元の濃度を位数といい、有限な位数を持つ体を有限体と呼び、そうでない体を無限体と呼ぶ。有限斜体は常に可換体である（ウェダーバーンの小定理）。

n・1 で単位元 1 を n 回足したものを表すとき、n・1 = 0 となるような正の整数 n のうち最も小さなものをその体の標数という。ただし、そのような n が存在しないとき標数は 0 であると決める。体の標数は 0 または素数である。

さて、英語版
”Division rings used to be called "fields" in an older usage. In many languages, a word meaning "body" is used for division rings, in some languages designating either commutative or non-commutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). ”
だって
つまり、英語では、"fields" は可換体で、 a word meaning "body"＝ 体は、”is used for division rings”だって。今の日本は、大分英語的用法になっているんだね（＾＾

https://en.wikipedia.org/wiki/Division_ring
Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring[1] in which every nonzero element a has a multiplicative inverse, i.e., an element x with a・x = x・a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. A division ring is a type of noncommutative ring under the looser definition where noncommutative ring refers to rings which are not necessarily commutative.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/449

450: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/20(木) 00:25:15 ID:gmO23IhH

>>449
つづき

Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".[5]

All division rings are simple, i.e. have no two-sided ideal besides the zero ideal and itself.

Main theorems
Wedderburn's little theorem: All finite division rings are commutative and therefore finite fields. (Ernst Witt gave a simple proof.)
Frobenius theorem: The only finite-dimensional associative division algebras over the reals are the reals themselves, the complex numbers, and the quaternions.

Related notions
Division rings used to be called "fields" in an older usage. In many languages, a word meaning "body" is used for division rings, in some languages designating either commutative or non-commutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). A more complete comparison is found in the article Field (mathematics).

The name "Skew field" has an interesting semantic feature: a modifier (here "skew") widens the scope of the base term (here "field"). Thus a field is a particular type of skew field, and not all skew fields are fields.

While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest.

A near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1595166668/450

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