\section{Real Numbers form Field}
Tags: Examples of Fields, Real Numbers

\begin{theorem}
The [[Definition:Real Number|set of real numbers]] $\R$ forms a [[Definition:Field (Abstract Algebra)|field]] under [[Definition:Real Addition|addition]] and [[Definition:Real Multiplication|multiplication]]: $\struct {\R, +, \times}$.
\end{theorem}

\begin{proof}
From [[Real Numbers under Addition form Abelian Group]], we have that $\struct {\R, +}$ forms an [[Definition:Abelian Group|abelian group]].
From [[Non-Zero Real Numbers under Multiplication form Abelian Group]], we have that $\struct {\R_{\ne 0}, \times}$ forms an [[Definition:Abelian Group|abelian group]].
Next we have that [[Real Multiplication Distributes over Addition]].
Thus all the criteria are fulfilled, and $\struct {\R, +, \times}$ is a [[Definition:Field (Abstract Algebra)|field]].
{{qed}}
\end{proof}


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