Henselian field

Remarks

Example: for a prime $p$, the field $\Q_p$ with the $p$-adic valuation is Henselian; this is Hensel's lemma in its $p$-adic form. More generally, a field that is complete for a valuation with values in $\R$ is Henselian. Non-example: $\Q$ with the $7$-adic valuation $v_7$ is not Henselian. Take $f = X^2 - 2$ and $a = 3$: $f(3) = 7 \in \mathfrak{m}$ and $f'(3) = 6 \notin \mathfrak{m}$, but $f$ has no root in $\Q$. Equivalent forms: $(K, v)$ is Henselian if and only if $\mathcal{O}$ is a Henselian ring. It is Henselian if and only if $v$ extends to a unique valuation on each algebraic extension of $K$. The element $b$ is then unique. Every field with the trivial valuation is Henselian.