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One of the first non-trivial results given in most courses on algebraic topology is the proof of the Fundamental Theorem of Algebra using topological methods. This is on page 11 of J.P. May's A Concise Course in Algebraic Topology, page 31 of Allen Hatcher's Algebraic Topology, page 65 of Albrecht Dold's Lectures on Algebraic Topology. This shows that a result which, developed classically, requires a fair bit of work and development, but when viewed with tools of a different field are easier and/or simpler. Are there any other notable examples of such situations?

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Any two distinct people at a party have exactly one common friend. Show that somebody is friend with everybody. - This looks like graph theory and combinatorics, but is very simple when using matrices and eigenvalues. – Hagen von Eitzen Jan 23 '13 at 15:33
There are a lot of examples of real variable results that (at present) have much simpler proofs using complex variable methods (certain definite integral evaluations, the prime number theorem, solving cubic algebraic equations with all real roots, etc.). Indeed, there is this well known (variously worded) quote by Jacques Hadamard: The shortest path between two truths in the real domain passes through the complex domain. – Dave L. Renfro Jan 23 '13 at 16:49

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