Prove {1,sqrt(2),sqrt(3)} are linearly independent over the rationals.?

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Prove {1,sqrt(2),sqrt(3)} are linearly independent over the rationals.?

I'm stuck on this proof. :(

2007-02-22 10:44:20 · 1 answers · asked by Tony O 2 in Science & Mathematics ➔ Mathematics

1 answers

Suppose the contrary. Then there exist rational numbers a and b such that

sqrt3 = a*1 + b*sqrt2

Square both sides...

3 = a² + 2ab*sqrt2 + 2

→ sqrt2 = (1 - a²) / 2ab

LHS is irrational, RHS is rational... contradiction.

2007-02-22 10:48:54 · answer #1 · answered by Anonymous · 1⤊ 0⤋