$ℤ/pℤ$的代数闭是无限域

hbghlyj

Page 10 M3P11, lectures by Kevin Buzzard Fall 2015

Examples (characteristic $p$ fields that are not $ℤ/pℤ$).
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There are also infinite examples.
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Another example would be the algebraic closure of $ℤ/pℤ$

我不懂得the algebraic closure of $ℤ/pℤ$为什么是无限的

hbghlyj

there are no finite algebraically closed fields means that the algebraic closure of a field of characteristic p will have to be an infinite field of characteristic p.

Czhang271828

hbghlyj 发表于 2023-8-10 22:04
there are no finite algebraically closed fields means that the algebraic closure of a field of characteristic $p$ will have to be an infinite field of characteristic $p$.

看了下 MSE 回答, 有个挺直接的观点没提到. "代数闭域"顾名思义, 即不存在其上的代数扩张; 但有限域上显然有有限扩张, 矛盾.