Fields

Overview

Let $R$ be a ring with unity $1\ne 0$. If every nonzero element of $R$ is a unit, then $R$ is a skew field (or division ring). $R$ is a field if it is a commutative division ring. $R$ is a strictly skew field if $R$ is a noncommutative division ring.

Subfields

Let $⟨F,+,\cdot ⟩$ be a field. Then $K$ is a subfield of $F$ if:

• $K$ is a subring of $F$ and
• every nonzero element of $K$ is a unit.

Homomorphisms

Let $F$ and $F^{\prime }$ be fields. A field homomorphism $\varphi :F\to F^{\prime }$ is a mapping that satisfies the following conditions for all $a,b\in F$:

• $\varphi (a+b)=\varphi (a){+}^{\prime }\varphi (b)$
• $\varphi (a\ast b)=\varphi (a){\ast }^{\prime }\varphi (b)$
• $\varphi (1)=1^{\prime }$

Note that $\varphi$ is also a ring homomorphism so the linked structural properties are preserved. Additionally,

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Special Cases

Rationals

The set of all rationals is denoted $\mathbb{Q}$. The following notation is often used to denote particular subsets of $\mathbb{Q}$:

• ${\mathbb{Q}}^{+}=\{q\in \mathbb{Q}\mid q>0\}$

Reals

The set of all real numbers is denoted $\mathbb{R}$. The following notation is often used to denote particular subsets of $\mathbb{R}$:

• ${\mathbb{R}}_c=\{x\in \mathbb{R}\mid 0\le x<c\}$ for $c\in \mathbb{R}$.
• ${\mathbb{R}}^{+}=\{x\in \mathbb{R}\mid x>0\}$.
• ${\mathbb{R}}^{\ast }=\{x\in \mathbb{R}\mid x\ne 0\}$.

Complex

The set of all complex numbers is a field, denoted $\mathbb{C}$. It is defined as

where $i$ is the imaginary number given by $i^2=-1$. Addition, subtraction, and multiplication are performed in the normal way. Division works with the aid of conjugates.

For any complex number of the form $z=a+bi$, $a,b\in \mathbb{R}$, the number $a$ is called the real part of $z$ and the number $b$ is called the imaginary part of $z$. This is denoted as $\text{Re}(z)=a$ and $\text{Im}(z)=b$ respectively.

The modulus of $z$, denoted $|z|$, is defined as $|z|=\sqrt{a^2+b^2}$. The argument of $z$, denoted $\text{Arg}(z)$, is the angle $-\pi <\theta \le \pi$ between the complex number and the positive real axis.

Conjugates

Let $z=a+bi$ be a complex number. Then its complex conjugate, denoted as $\overline{z}$, is defined as

Arithmetic

The normal arithmetic operations hold in $\mathbb{C}$. If $z_1,z_2\in \mathbb{C}$ where $z_1=a+bi$ and $z_2=c+di$,

• Addition.$$(a+bi)+(c+di)=(a+c)+(b+d)i$$
• Subtraction.$$z_1-z_2=z_1+(-z_2)$$
• Multiplication.$$(a+bi)\cdot (c+di)=(ac-bd)+(ad+bc)i$$
• Division.$$\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{(c+di)(c-di)}=\frac{(a+bi)(c-di)}{c^2+d^2}.$$