91Ó°ÊÓ

Prove that cvery positive real number has a positive square root in \(R\). Since the polynomial \(t^{2}-a\) has at most two roots in a field, and since for any root \(\alpha\), the number \(-\alpha\) is also a root, it follows that for every \(a \in \mathbb{R}, a \geq 0\), there exists a unique \(\alpha \in \mathbf{R}, \alpha \geq 0\) such that \(\alpha^{2}=a .\) [Hint: For the above proof, let \(\alpha\) be the least upper bound of the set of rational numbers \(b\) such that \(b^{2} \leqq a\).]

Show that every automorphism of the real numbers is the identity. [Hint: Show first that an automorphism is order preserving.]

An absolute value is called non-archimedean if instead of AV 3 it satisfies the stronger property $$ |x+y| \leqq \max (|x|,|y|) $$ The function \(\mid I_{F}\) on \(Q\) is called the \(p\) -adic absolute value, and is nonarchimedean. Suppose | | is a non-archimedean absolute value on a field \(\boldsymbol{F}\). Prove that given \(x \in F, x \neq 0\), there exists a positive number \(r\) such that if \(|y-x|