Exercise 2.3.1 (Easy fields)

Question

Write down all the examples of fields that you know.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
$\mathbb{C}, \mathbb{R}, \mathbb{Q}$

Richard Ganaye · Answer

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

To $\Q, \R,\C$, we can add

$\F_p = \Z/p\Z$, where $p$ is a prime number,

$\R(x),\R(x,y), \F_p(x),\ldots$,

$\R[x]/\langle x^3 - 2 angle \simeq \R(\sqrt[3]{2})$

$\F_5[x]/\langle x^2 - 2 angle$,

the $p$-adic fields $\Q_p$,

$\ldots$

Answers