  We continue the study of $n$-dependent groups, fields and related structures,
largely motivated by the conjecture that every $n$-dependent field is
dependent. We provide evidence towards this conjecture by showing that every
infinite $n$-dependent valued field of positive characteristic is henselian,
obtaining a variant of Shelah's Henselianity Conjecture in this case and
generalizing a recent result of Johnson for dependent fields. Additionally, we
prove a result on intersections of type-definable connected components over
generic sets of parameters in $n$-dependent groups, generalizing Shelah's
absoluteness of $G^{00}$ in dependent theories and relative absoluteness of
$G^{00}$ in $2$-dependent theories. In an effort to clarify the scope of this
conjecture, we provide new examples of strictly $2$-dependent fields with
additional structure, showing that Granger's examples of non-degenerate
bilinear forms over dependent fields are $2$-dependent. Along the way, we
obtain some purely model-theoretic results of independent interest: we show
that $n$-dependence is witnessed by formulas with all but one variable
singletons; provide a type-counting criterion for $2$-dependence and use it to
deduce $2$-dependence for compositions of dependent relations with arbitrary
binary functions (the Composition Lemma); and show that an expansion of a
geometric theory $T$ by a generic predicate is dependent if and only if it is
$n$-dependent for some $n$, if and only if the algebraic closure in $T$ is
disintegrated. An appendix by Martin Bays provides an explicit isomorphism in
the Kaplan-Scanlon-Wagner theorem.

Logic;Commutative Algebra;Group Theory