# Existence

# Existence

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras *D*₁, *D*₂, and *D*₃ over *K* containing a common subfield *E* such that *E*/*K* is a ramified separable extension.

*Keywords:*
quadratic form, Bruhat-Tits building, Pfister form, Structure Theorem, unramified quadratic space, unramified separable quadratic extension, tamely ramified division algebra, ramified separable quadratic extension, unramified quaternion division algebra, ramified quaternion division algebra

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