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  "body": "\n## Definition\n\nLet $K$ be a field that is complete with respect to a nontrivial [[non-archimedean absolute value|1H5F1BP6XS]] $\\abs{\\cdot}$, and let $n \\ge 0$.\nFor a multi-index $\\nu = (\\nu_1, \\dots, \\nu_n) \\in \\N^n$ write $X^\\nu = X_1^{\\nu_1} \\cdots X_n^{\\nu_n}$ and $\\abs{\\nu} = \\nu_1 + \\cdots + \\nu_n$.\nThe *Tate algebra* in $n$ variables over $K$ is the subring\n$$T_n = K\\langle X_1, \\dots, X_n \\rangle = \\Big\\{ \\sum_{\\nu \\in \\N^n} a_\\nu X^\\nu \\in K[\\![X_1, \\dots, X_n]\\!] : \\abs{a_\\nu} \\to 0 \\text{ as } \\abs{\\nu} \\to \\infty \\Big\\}$$\nof the [[formal power series|033WGN00BF]] ring.\nIts *Gauss norm* is $\\norm{\\sum a_\\nu X^\\nu} = \\max_\\nu \\abs{a_\\nu}$.\n\n## Remarks\n\nThe maximum exists because $\\abs{a_\\nu} \\to 0$.\n$T_n$ is closed under multiplication, and the Gauss norm is multiplicative: $\\norm{fg} = \\norm{f} \\norm{g}$.\nWith this norm $T_n$ is a [[Banach algebra|9KYWN94QDN]] over $K$, and the polynomial ring $K[X_1, \\dots, X_n]$ is dense in it.\nLet $\\bar K$ be an algebraic closure of $K$ with the extended absolute value.\nA formal power series lies in $T_n$ exactly when it converges at every point $x \\in \\bar K^n$ with all $\\abs{x_i} \\le 1$.\n$T_n$ is Noetherian, regular and a unique factorization domain, and it satisfies a Noether normalization lemma.\nThe residue field of each maximal ideal of $T_n$ is a finite extension of $K$.\n\nExample: over the [[completion|W168DS966V]] $\\Q_p$ of $\\Q$ for the $p$-adic absolute value, $f = \\sum_{k \\ge 0} p^k X^k$ lies in $T_1$, because $\\abs{p^k}_p = p^{-k} \\to 0$; its Gauss norm is $\\abs{1}_p = 1$.\n\nNon-example: $\\sum_{k \\ge 0} X^k$ has all coefficients of absolute value $1$, which do not tend to $0$, so it is not in $T_1$; it diverges at $X = 1$.\n",
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