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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a [[vector space|000AYPA9AJ]] over $F$, and let $T(V)$ be its [[tensor algebra|P4J5JZDW0B]].\nLet $I$ be the [[two-sided ideal|0CD87SFM2N]] of $T(V)$ generated by the elements $v \\otimes w - w \\otimes v$ with $v, w \\in V$.\nThe *symmetric algebra* of $V$ is the [[quotient ring|0076HWHGHQ]]\n$$S(V) = T(V) / I.$$\n\n## Remarks\n\n$S(V)$ is a commutative associative unital $F$-algebra, graded by degree; its degree $k$ part $S^k(V)$ is the $k$-th symmetric power of $V$.\nUniversal property: every linear map from $V$ to a commutative associative unital $F$-algebra $A$ extends uniquely to a unital algebra homomorphism $S(V) \\to A$.\nExample: if $V$ has basis $x_1, \\ldots, x_n$, then $S(V) \\cong F[x_1, \\ldots, x_n]$, and $S^k(V)$ has dimension $\\binom{n + k - 1}{k}$; for $n = 2$, $k = 2$ the basis is $x_1^2, x_1 x_2, x_2^2$.\nNon-example: for $\\dim V \\ge 2$, the tensor algebra $T(V)$ itself is not commutative, so it is not $S(V)$.\nThe [[exterior algebra|0MDMZ965VE]] uses instead the ideal generated by the elements $v \\otimes v$.\n",
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