## Fields

• A field is a set $𝔽$ with functions  $\begin{array}{ccc}𝔽×𝔽& \to & 𝔽\\ \left({c}_{1},{c}_{2}\right)& ⟼& {c}_{1}+{c}_{2}\end{array}$     and     $\begin{array}{ccc}𝔽×𝔽& \to & 𝔽\\ \left({c}_{1},{c}_{2}\right)& ⟼& {c}_{1}{c}_{2}\end{array}$,
(a)   If ${c}_{1},{c}_{2},{c}_{3}\in 𝔽$ then $\left({c}_{1}+{c}_{2}\right)+{c}_{3}=\left({c}_{1}+\left({c}_{2}+{c}_{3}\right)$,
(b)   If ${c}_{1},{c}_{2}\in 𝔽$ then ${c}_{1}+{c}_{2}={c}_{2}+{c}_{1}$,
(c)   There exists a zero, $0\in 𝔽$, such that if $c\in 𝔽$ then $0+c=c+0=c$,
(d)   If $c\in 𝔽$ then there exists an additive inverse to $c$, $-c\in 𝔽$, such that $\left(-c\right)+c=c+\left(-c\right)=0$,
(e)   If ${c}_{1},{c}_{2},{c}_{3}\in 𝔽$ then ${c}_{1}\left({c}_{2}{c}_{3}\right)=\left({c}_{1}{c}_{2}\right){c}_{3}$,
(f)   If ${c}_{1},{c}_{2}\in 𝔽$ then ${c}_{1}{c}_{2}={c}_{2}{c}_{1}$,
(g)   There exists an identity, $1\in 𝔽$, such that $1\ne 0$ and if $c\in 𝔽$ then $1c=c1=c$,
(h)   If $c\in 𝔽$ and $c\ne 0$ then there exists an inverse (sometimes called a multiplicative inverse), ${c}^{-1}\in 𝔽$, such that $c{c}^{-1}={c}^{-1}c=1$,
(i)   If ${c}_{1},{c}_{2},{c}_{3}\in 𝔽$ then ${c}_{1}\left({c}_{2}+{c}_{3}\right)={c}_{1}{c}_{2}+{c}_{1}{c}_{3}$.
• A subfield of a field $𝔽$ is a subset $𝕂\subseteq 𝔽$ such that
(a)   If ${k}_{1},{k}_{2}\in 𝕂$ then ${k}_{1}+{k}_{2}\in 𝕂$,
(b)   $0\in 𝕂$,
(c)   If $k\in 𝕂$ then $-k\in 𝕂$,
(d)   If ${k}_{1},{k}_{2}\in 𝕂$ then ${k}_{1}{k}_{2}\in 𝕂$,
(e)   $1\in 𝕂$,
(f)   If $k\in 𝕂$ then ${k}^{-1}\in 𝕂$,

Note that every field is a commutative ring and the only conditions in the defnition of a field that are not in the definition of a ring are (f) and (h).

Important examples of fields are:

(a)   The rational numbers $ℚ$, the real numbers $ℝ$, and the complex numbers $ℂ$,
(b)   $ℤ/pℤ$, where $p$ is a prime.

#### Homomorphisms

Field homomorphisms might be used to compare fields. The only problem is that there aren't many interesting field homomorphisms, as we show in Proposition (fldhominj). We shall study fields in more depth in Part V.

Let $𝕂$ and $𝔽$ be fields with identities ${1}_{𝕂}$ and ${1}_{𝔽}$, respectively.

• A field homomorphism from $𝕂$ to $𝔽$ is a function $f:𝕂\to 𝔽$ such that
(a)   If ${k}_{1},{k}_{2}\in 𝕂$ then $f\left({k}_{1}+{k}_{2}\right)=f\left({k}_{1}\right)+f\left({k}_{2}\right)$,
(b)   If ${k}_{1},{k}_{2}\in 𝕂$ then $f\left({k}_{1}{k}_{2}\right)=f\left({k}_{1}\right)f\left({k}_{2}\right)$,
(c)   $f\left({1}_{𝕂}\right)=f\left({1}_{𝔽}\right)$.

HW: Show that if $f:𝕂\to 𝔽$ is a field homomorphism then $f\left({0}_{𝕂}\right)={0}_{𝔽}$, where ${0}_{𝕂}$ and ${0}_{𝔽}$ are the zeros in $𝕂$ and $𝔽$, respectively.
HW: Explain why conditions (a) and (b) in the definition of a field homomorphism do not imply condition (c).

If $f:𝕂\to 𝔽$ is a field homomorphism then $f$ is injective.

Proposition (fldhominj) stated another way, says that the kernel of any field homomorphism is $\left\{0\right\}$. This means that we cannot get an interesting analogue of Theorem ??? for fields. Proposition (fldhominj) also shows that if $f:𝕂\to 𝔽$ is a field homomorphism then $\mathrm{im}f=f\left(𝕂\right)$ is a subfield of $𝔽$.

## Notes and References

These notes are written to highlight the analogy between groups and group actions, rings and modules, and fields and vector spaces.

## References

[Ram] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1993-1994.

[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.