More Notes on Galois Theory
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چکیده
Despite its name, the Fundamental Theorem of Algebra cannot be a result in pure algebra since the real numbers and hence the complex numbers are not algebraically defined. While there are many proofs, most use some basic facts in complex analysis or plane topology. We describe here a proof based on Galois theory as well as some non-trivial finite group theory, namely the Sylow theorems, but which only depends on two basic facts about the real numbers R which are usually presented (but not carefully proved!) in a first year calculus class, both of which depend on the Intermediate Value Theorem:
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