#### Vol. 292, No. 2, 2018

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Norms in central simple algebras

### Daniel Goldstein and Murray Schacher

Vol. 292 (2018), No. 2, 373–388
##### Abstract

Let $A$ be a central simple algebra over a number field $K$. We study the question of which integers of $K$ are reduced norms of integers of $A$. We prove that if $K$ contains an integer that is the reduced norm of an element of $A$ but not the reduced norm of an integer of $A$, then $A$ is a totally definite quaternion algebra over a totally real field (i.e., $A$ fails the Eichler condition).

 To Robert Steinberg, a cherished teacher, colleague, and friend
##### Keywords
central simple algebras, reduced norms, super singular elliptic curves
Primary: 11S45
Secondary: 11S15