# Smallest Field is Field

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Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: Amend this, or add another proof, to use the ring of integers modulo $2$ or something. It's currently unwieldy. We need to show a) that this is a field, and b) that this is the smallest field.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Theorem

The ring $\struct {\set {0_R, 1_R}, +, \circ}$ is the smallest algebraic structure which is a field.

## Proof

From Field Contains at least 2 Elements, a field must contain at least two elements.

Hence the null ring, which contains one element, is *not* a field.

For $\struct {\set {0_R, 1_R}, +, \circ}$ to be a field:

- $\struct {\set {0_R, 1_R}, +}$ must be an abelian group.

This is fulfilled as this is the parity group.

- $\struct {\set {0_R, 1_R}, \circ}$ must be a commutative division ring.

This is fulfilled, as $\struct {\set {0_R, 1_R}^*, \circ} = \struct {\set {1_R}, \circ}$ is the trivial group.

- $\circ$ needs to distribute over $+$. This follows directly from Ring Product with Zero and the behaviour of the identity element in a group.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 5$. Further Examples of Integral Domains: Example $6$ (which proves the long way round that this is an integral domain) - 1973: C.R.J. Clapham:
*Introduction to Mathematical Analysis*... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $4$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**field**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**field**:**1.**