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Homework Statement
Homework Equations
The Attempt at a Solution
For part (a):
a*1 + b*√2 + c*√3 = 0
assume a, b, c not all zero
a + b√2 = c√3
a^{2} + 2b^{2} + 2ab√2 = 3c^{2}
a^{2} + 2b^{2}  3c^{2} = 2ab√2
(a^{2} + 2b^{2}  3c^{2})/(2ab) = √2
which is not possible since we take a, b, c to be rational, and √2 is irrational.
thus our assumption of a, b, c not all zero was false and we must have a=b=c=0.
For part (b), a similar argument, but easier:
{1, 1 + √5, (1 + √5)^{2}} = {1, 1 + √5, 1 + 2√5 + 5}
a + b(1 + √5) + c(1 + 2√5 + 5) = 0
assume a, b, c not all zero
1 + b + b√5 + c + 2c√5 + 5c = 0
1 + b + c + 5c = b√5  2c√5
1 + b + 6c = (b  2c)√5
(1 + b + 6c)/(b  2c) = √5
same story as before.
Now that was super easy. And the assignment says to be careful with the structure of my argument. And I hate denominators because they can't be zero. If a = b = c = 0, then we have 0/0, which is of indeterminate form, which is ok! but i'm getting the idea that my argument is flawed because of division by zero...
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