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  "body": "\n## Statement\n\nLet $G$ be a group and let $L$ be a field, with multiplicative [[group of units|05THZKR9AN]] $L^\\times$.\nLet $\\chi_1, \\ldots, \\chi_n \\colon G \\to L^\\times$ be pairwise distinct [[group homomorphisms|0HV9Y7WG5R]].\nThen $\\chi_1, \\ldots, \\chi_n$ are [[linearly independent|0EVHJFTG9Z]] over $L$ as functions $G \\to L$: if $a_1, \\ldots, a_n \\in L$ and $a_1 \\chi_1(g) + \\cdots + a_n \\chi_n(g) = 0$ for all $g \\in G$, then $a_1 = \\cdots = a_n = 0$.\n\n## Remarks\n\nSuppose there is a nontrivial relation, and take one $\\sum a_i \\chi_i = 0$ with the fewest nonzero coefficients; then at least two coefficients, say $a_1$ and $a_2$, are nonzero.\nChoose $h \\in G$ with $\\chi_1(h) \\ne \\chi_2(h)$.\nEvaluating the relation at $hg$ and subtracting $\\chi_1(h)$ times the relation at $g$ gives $\\sum_i a_i (\\chi_i(h) - \\chi_1(h)) \\chi_i = 0$.\nThis relation has fewer nonzero coefficients but a nonzero coefficient at $\\chi_2$, a contradiction.\n\nThe main application concerns [[field homomorphisms|EZVN87C88J]].\nIf $\\sigma_1, \\ldots, \\sigma_n \\colon K \\to L$ are distinct field homomorphisms, their restrictions to the group $K^\\times$ are distinct characters, so $\\sigma_1, \\ldots, \\sigma_n$ are linearly independent over $L$.\nFor example, the identity and complex conjugation on $\\C$ are independent over $\\C$: if $a z + b \\bar z = 0$ for all $z$, then $z = 1$ gives $a + b = 0$ and $z = i$ gives $a - b = 0$, so $a = b = 0$.\n\nThe same proof works when $G$ is only a monoid and the $\\chi_i$ are monoid homomorphisms into $L^\\times$.\n\n## Explanations\n\n### L1 human\n\nA character is a map from a group to the nonzero numbers of a field that turns products into products.\nDedekind's lemma says that different characters are genuinely different as functions: no one of them is a combination of the others.\nIt is the field-theoretic cousin of the fact that the functions $e^{a_1 x}, \\ldots, e^{a_n x}$ with distinct $a_i$ are linearly independent.\n\n### L2 human\n\nIn Galois theory the lemma is used to show that a finite extension $L/K$ has at most $[L:K]$ automorphisms over $K$.\nIt is the key step in the proofs of [[Artin's theorem on fixed fields|X7R15C00ET]], of [[Hilbert's Theorem 90|YFF8B5A0QG]] and of the [[normal basis theorem|PXY1WYNRD4]].\nIt also shows that the [[field trace|WPE5Z53QYJ]] of a finite separable extension is not identically zero.\n",
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