Field axioms

Field axioms are the rules a field satisfies.^[1]^[2]

Field Axioms Media

The regular heptagon cannot be constructed using only a straightedge and compass construction; this can be proven using the field of constructible numbers.
The multiplication of complex numbers can be visualized geometrically by rotations and scalings.
The geometric mean theorem asserts that $h 2 = pq$ . Choosing $q = 1$ allows construction of the square root of a given constructible number $p$ .
In modular arithmetic modulo 12, 9 + 4 = 1 since 9 + 4 = 13 in $Z$ , which divided by 12 leaves remainder 1. However, $Z /12 Z$ is not a field because 12 is not a prime number.
Each bounded real set has a least upper bound.
The sum of three points $P$ , $Q$ , and $R$ on an elliptic curve $E$ (red) is zero if there is a line (blue) passing through these points.
A compact Riemann surface of genus two (two handles). The genus can be read off the field of meromorphic functions on the surface.
The fifth roots of unity form a regular pentagon.
The hairy ball theorem states that a ball cannot be combed. More formally, there is no continuous tangent vector field on the sphere $S2$ , which is everywhere non-zero.

↑ Weisstein, Eric W. "Field Axioms". mathworld.wolfram.com. Retrieved 2021-09-15.
↑ Apostol, T. M. "The Field Axioms." §I 3.2 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 17-19, 1967.