斜体 - あくまで個人的な日記であって

大体、午前に非可換体(斜体)などと書いたが、ハミルトンの四元数体Hぐらいしか知らない。

実数体R上の有限次元associative代数で体になるのはR,C(複素数体),Hしかない(Frobenius)

ということは知っているが、非可換な体がどれだけあるのか知らない。

The subset of the quaternions $a + bi + cj + dk$ , such that $a$ , $b$ , $c$ , and $d$ belong to a fixed subfield of the real numbers, is a noncommutative division ring. When this subfield is the field of rational numbers, this is the division ring of rational quaternions.

こっちはすぐ想像できる。だから構造が易しいとは言わない。

で、

Let $\sigma :\mathbb {C} \rightarrow \mathbb {C}$ be an automorphism of the field $\mathbb {C}$ . Let $\mathbb {C} *2$ denote the ring of formal Laurent series with complex coefficients, wherein multiplication is defined as follows: instead of simply allowing coefficients to commute directly with the indeterminate $z$ , for $\alpha \in \mathbb {C}$ , define $z^{i}\alpha :=\sigma ^{i}(\alpha )z^{i}$ for each index $i\in \mathbb {Z}$ . If $\sigma$ is a non-trivial automorphism of complex numbers (such as the conjugation), then the resulting ring of Laurent series is a strictly noncommutative division ring known as a skew Laurent series ring;^[9] if $σ = id$ then it features the standard multiplication of formal series. This concept can be generalized to the ring of Laurent series over any fixed field $F$ , given a nontrivial $F$ -automorphism $\sigma$ .