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  "body": "\n## Summary\n\nThe Tate algebra in $n$ variables over a complete non-archimedean field $K$ is the algebra of power series whose coefficients tend to $0$.\nThese are exactly the power series that converge on the closed unit polydisc over an algebraic closure of $K$.\nWith the Gauss norm it is a [[Banach algebra|9KYWN94QDN]], and it is a [[Noetherian|0670T59HGE]] unique factorization domain.\nIts quotients, the affinoid algebras, are the local building blocks of [[rigid analytic spaces|YAH6MZVG7K]], as polynomial rings are for algebraic varieties.\n\n## Explanations\n\n### L1 human\n\nOver the [[$p$-adic numbers|BT6T702C28]], a series $\\sum a_k$ converges exactly when its terms tend to $0$.\nSo a power series $\\sum a_k x^k$ converges at every $x$ with $\\abs{x} \\le 1$ exactly when $\\abs{a_k} \\to 0$.\nThe Tate algebra collects these power series, in one or several variables.\nFor example $\\sum_k p^k x^k$ belongs to it, while $\\sum_k x^k$ does not, since it diverges at $x = 1$.\nIt plays the role that the ring of holomorphic functions on a closed polydisc plays in complex analysis.\n",
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