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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]], let $n \\ge 1$, and let $M_n(k)$ be the ring of $n \\times n$ [[matrices|0JXZKX526R]] over $k$.\nLet $f \\in k\\langle x_1, \\ldots, x_m \\rangle$ be an element of the [[free algebra|2P3Y2FRCSZ]] with zero constant term.\nThe polynomial $f$ is a *central polynomial* for $M_n(k)$ if:\n\n1. $f(a_1, \\ldots, a_m)$ is a scalar matrix $\\lambda I$, with $\\lambda \\in k$, for all $a_1, \\ldots, a_m \\in M_n(k)$;\n2. $f$ is not a [[polynomial identity|C0EDT81EGH]] of $M_n(k)$: $f(a_1, \\ldots, a_m) \\ne 0$ for some $a_1, \\ldots, a_m \\in M_n(k)$.\n\n## Remarks\n\nClause 1 says that all values of $f$ lie in the [[center|MVT64B3B6H]] $k I$ of $M_n(k)$.\n\nExample: $f = (x_1 x_2 - x_2 x_1)^2$ is a central polynomial for $M_2(k)$.\nFor $a, b \\in M_2(k)$ the matrix $c = ab - ba$ has trace $0$, so the [[Cayley–Hamilton theorem|0F765KFC2J]] gives $c^2 = -\\det(c) \\, I$, a scalar matrix.\nFor $a = e_{12}$ and $b = e_{21}$ one has $c = e_{11} - e_{22}$ and $c^2 = I \\ne 0$, so clause 2 holds.\n\nNon-example: $g = x_1 x_2 - x_2 x_1$ fails clause 1 for $M_2(k)$, since $g(e_{12}, e_{21}) = e_{11} - e_{22}$ is not scalar.\nNon-example: $s_4$, the standard polynomial of degree $4$, satisfies clause 1 for $M_2(k)$ only because all its values are $0$, so it fails clause 2.\n\nFor $n = 1$ every polynomial with a nonzero value on $k$ is central, so the notion is of interest for $n \\ge 2$.\nCentral polynomials exist for every $n \\ge 1$, a result of Formanek and of Razmyslov.\n",
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