Question

Prove that 1, √ 2, √ 3, and √ 6 are linearly independent over Q. Hint:...

Prove that 1, √ 2, √ 3, and √ 6 are linearly independent over Q.

Hint: Suppose that p + q √ 2 + r √ 3 + s √ 6 = 0 with p, q, r, s ∈ Q. We may suppose that r 6= 0 or s 6= 0 (why?).

If so, then we can write √ 3 in the form √ 3 = (a + b √ 2) / (c + d √ 2) = e + f √ 2

where a, b, c, d, e, f ∈ Q. Square both sides and obtain a contradiction

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Answer #1

I think this proof is more simple an easier than the hint given . It is so simple proof that any one can understand it . Hopefully you understand this proof . If you still have any doubt please let me know in comments. I shall reply you as soon as possible. Thank you.

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