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We discuss and relate finiteness conditions for certain field invariants which are studied in quadratic form theory. This includes the $u$-invariant, the reduced stability index and the symbol lengths for Galois cohomology groups with coefficients in $\mu_2=\{+1,-1\}$, as well as a new invariant called the splitting height.

Let $F$ be the function field of a curve over a non-archimedean local field. Let $m \geq 2$ be an integer coprime to the characteristic of the residue field of the local field. In this article, we show that every element in $H^{3}(F, \mu_{m}^{\otimes 2})$ is of the form $\chi \cup (f) \cup (g)$, where $\chi$ is in $H^{1}(F, \mathbb{Z}/m\mathbb{Z})$ and $(f)$, $(g)$ in $H^{1}(F, \mu_{m})$. This extends a result of Parimala and Suresh, where they show this when $m$ is prime and when $F$ contains $\mu_{m}$.

Let $F$ be the function field of a curve over a complete discretely valued field. Let $\ell$ be a prime not equal to the characteristic of the residue field. Given a finite subgroup $B$ in the $\ell$ torsion part of the Brauer group ${}_{\ell}Br(F)$, we define the index of $B$ as the minimum of the degrees of field extensions which split all elements in $B$. In this manuscript, we give an upper bound for the index of any finite subgroup $B$ in terms of arithmetic invariants of $F$. As a simple application of our result, given a quadratic form $q/F$, where $F$ is the function field of a curve over an $n$-local field, we provide an upper bound to the minimum of degrees of field extensions $L/F$ so that the Witt index of $q\otimes L$ becomes the largest possible., Comment: Proposition 3.1 and Corollary 3.2 now generalized. Removed Remark 3.4. Other stylistic changes. Comments welcome

We study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements. We show in particular that for all $m,n \in \mathbb{Z}^+$ with $n \geq 3$ and $4 \leq m \leq \frac{n}{3}$ there is an atomic domain with precisely $n$ atoms, precisely $m$ maximal ideals and no prime elements. The proofs use both commutative algebra and additive number theory., Comment: 13 pages