Example: Examples of Fields

(related to Chapter: Fields (Overview))

Below, you can find some examples of the algebraic structure called a field.

Example 1

The rational numbers together with the addition "$+$" and multiplication "$\cdot$", i.e. the algebraic structure $(\mathbb Q,+,\cdot)$ form a field, which is proven here.

Counterexample 2

The integers $(\mathbb Z, +,\cdot)$ are not a field since $1$ and $-1$ are the only integers that have a multiplicative inverse. To be a field, it would be necessary that all elements except the additive neutral element $0$ had a multiplicative inverse.

Counterexample 3

The set of all $n\times n$ matrices over a field $F$ together with matrix addition and matrix multiplication an $(M_{n\times n}(F), +,\cdot)$ are not a field, since not all such matrices unequal the zero matrix are invertible matrix multiplicative inverse. Also, the multiplication is in general not commutative.

Example 4

The real numbers $(\mathbb R, +,\cdot)$ form a field (was proven here).

Example 5

The complex numbers $(\mathbb C, +,\cdot)$ form a field (was proven here).

Example 6

The smallest finite field is the algebraic structure $(\{1,0\},+,\cdot)$ with the following addition and multiplication, which can be easily verified by the reader as an exercise (just verify all the field axioms):

$$\begin{array}{c|cc} +&1&0\\\hline 1&0&1\\ 0&1&0 \end{array}\quad\begin{array}{c|cc} \cdot &1&0\\\hline 1&0&0\\ 0&0&1 \end{array}$$

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  1. Modler, F.; Kreh, M.: "Tutorium Analysis 1 und Lineare Algebra 1", Springer Spektrum, 2018, 4th Edition