Absolute value on a field ($|\cdot |$)

An absolute value on a field is a nonnegative multiplicative function that vanishes only at zero and satisfies the triangle inequality.

A non-Archimedean absolute value satisfies the ultrametric inequality $|x+y|\le \max (|x|,|y|)$.

The ultrametric inequality is $d(x,z)\le \max (d(x,y),d(y,z))$. For unequal $|x|$ and $|y|$, it implies $|x+y|=\max (|x|,|y|)$.

An ultrametric space is a metric space satisfying the ultrametric inequality. Its open balls are clopen, and any two balls are either disjoint or one contains the other.

A topological space is totally disconnected when its only connected subsets are singletons.

Two nontrivial absolute values on a field are equivalent when they induce the same topology. Equivalently, one is a positive real power of the other.

For a field extension $L/K$, an extension of an absolute value $|\cdot |$ on $K$ is an absolute value on $L$ whose restriction to $K$ is $|\cdot |$.

If $L=K(\alpha )$ is a finite separable extension and $v$ is an absolute value on $K$, the extensions of $v$ to $L$ correspond to the irreducible factors of the minimal polynomial of $\alpha$ over the completion of $K$ at $v$.