Is an automorphism of the field $\mathbb{Q}_p$ of $p$-adic numbers the identity map? If yes, how can we prove it?
Note:We don't assume an automorphism of $\mathbb{Q}_p$ is continuous.
Is an automorphism of the field $\mathbb{Q}_p$ of $p$-adic numbers the identity map? If yes, how can we prove it?
Note:We don't assume an automorphism of $\mathbb{Q}_p$ is continuous.
We use squares in a way that resembles the usual proof of the corresponding result about reals - we use squares to prove continuity of automorphisms.
Assume first that $p>2$ and that $\sigma$ is an automorphism of $\mathbb{Q}_p$. The key observation is that $1+px^2$ is a square in $\mathbb{Q}_p$ if and only if $x$ is in $\mathbb{Z}_p$. If $x$ is not a $p$-adic integer, then $\nu(1+px^2)$ is odd, so it cannot be a square. On the other hand (this is where we need $p>2$) by Hensel's lemma $1+px^2$ is a square, if $x$ is a $p$-adic integer.
If $1+px^2=y^2$, then clearly $1+p\sigma(x)^2=\sigma(y)^2$, so from the preceding paragraph we can deduce that $\sigma(x)\in\mathbb{Z}_p$ whenever $x$ is. But $\sigma(p)=p$, so we see that $\sigma^{-1}(p^k\mathbb{Z}_p)=p^k\mathbb{Z}_p$ for all natural numbers $k$. Thus $\sigma$ is continuous, and the claim follows from density of $\mathbb{Z}$ (they are all fixed points of $\sigma$) inside $\mathbb{Z}_p$.
If $p=2$ we need to make a small modification to the above argument. This time we see that $1+8x^2$ is a $2$-adic square, iff $x$ is a $2$-adic integer. This is because Hensel's lemma allows us to prove the existence of a $2$-adic square root of anything $\equiv 1\pmod8$. On the other hand if $x\notin\mathbb{Z}_2$, then either $\nu(1+8x^2)$ is odd (whenever $\nu(x)\le -2$) or $1+8x^2$ is an odd $2$-adic integer $\not\equiv1\pmod4$. It cannot be a square in either case. The rest of the argument works as above.
The following proof is basically the same as Mr. Dietrich Burde's answer.
Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\nu$ be the canonical additive valuation on $\mathbb{Q}_p$, i.e. $\nu(p) = 1$.
Let $\sigma$ be an automorphism of $\mathbb{Q}_p$. Since $\mathbb{Q}$ is dense in $\mathbb{Q}_p$ and $\sigma$ induces the identity map on $\mathbb{Q}$, it suffices to prove that $\sigma$ is continuous.
Let $U$ be the set of $p$-adic units. We first show that $\sigma(U) \subset U$. Let $\alpha \neq 0$ be a $p$-adic number. Suppose the set $S = \{n \in \mathbb{Z}, n \gt 0\mid x^n = \alpha$ has a solution in $\mathbb{Q}_p\}$ is infinite. If $x^n = \alpha$, then $n\nu(x) = \nu(\alpha)$. Hence if $\nu(\alpha) \neq 0$, then $\nu(\alpha)$ is divisible by infinite numbers of rational integers. This is absurd. Therefore $\nu(\alpha) = 0$, which means $\alpha \in U$.
Now suppose $\epsilon$ is a $p$-adic unit. The set $S = \{n \in \mathbb{Z}, n \gt 0\mid x^n = \epsilon$ has a solution in $\mathbb{Q}_p\}$is infinite as shown in here. Hence the similar set for $\sigma(\epsilon)$ is infinite. Therefore $\sigma(\epsilon) \in U$ by what we have shown above. This means $\sigma(U) \subset U$.
Let $\alpha \neq 0$ be an element of $\mathbb{Z}_p$. Let $n = \nu(\alpha)$. Then $\alpha = p^n \epsilon$, where $\epsilon \in U$. Since $\sigma(\alpha) = p^n\sigma(\epsilon)$, $\sigma(\alpha) \in \mathbb{Z}_p$. Hence $\sigma(\mathbb{Z}_p) \subset \mathbb{Z}_p$. Hence $\sigma(p^n\mathbb{Z}_p) \subset p^n\mathbb{Z}_p$ for every positive integer $n$. This means $\sigma$ is continuous.
The identity map is the only automorphism of the field of $p$-adic numbers, because of Schmidt's theorem implying that a field complete with respect to a discrete absolute value is not complete with respect to an absolute value, which is inequivalent to the original. For a proof, see here: http://www.math.utk.edu/~wagner/papers/padic.pdf