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  "body": "\n## Summary\n\nA central simple algebra over a field $K$ is a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$ that is a [[simple ring|62M0NC3E2T]].\nIts [[center|MVT64B3B6H]] must be exactly $K$.\nBy Wedderburn's theorem each one is a matrix algebra $M_n(D)$ over a central division algebra $D$.\nThey are stable under tensor product over $K$, and their classes up to matrix size form the [[Brauer group|BEW48TA9WZ]] of $K$.\nThey are central in algebraic number theory, in Galois cohomology and in the arithmetic of quadratic forms.\n\n## Explanations\n\n### L1 human\n\nThe basic example is the algebra of $n \\times n$ matrices over a field $K$.\nIt is simple, since its only two-sided ideals are $0$ and itself, and its center consists of the scalar matrices, which form a copy of $K$.\nCentral simple algebras are the algebras that look like matrix algebras after the field is enlarged.\nFor example, Hamilton's [[quaternions|0GE4DVQKR8]] are central simple over $\\R$, and over $\\C$ they become the $2 \\times 2$ complex matrices.\n",
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