THEOREM OF THE DAY: The Lindemann-Weierstrass Theorem

Theorem

If ${\alpha }_1,...,{\alpha }_n,n\ge 1$, are algebraic numbers which are linearly independent over $Q$, then $e^{{\alpha }_1},...,e^{{\alpha }_n}$ are algebraically independent; that is, any rational polynomial $P(z_1,...,z_n)$, having algebraic coefficients, for which $P(e^{{\alpha }_1},...,e^{{\alpha }_n})=0$, must be identically zero.

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