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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