Prove that 1 Squareroot2 Squareroot3is linearly independent

Prove that {1, Squareroot2, Squareroot3}is linearly independent in the Q-vector space R.

Solution

This is equivalent to showing that if

a+b6=c3+d2a+b6=c3+d2

then a=b=c=d=0a=b=c=d=0.

By squaring both sides of the above equation we get

a2+6b2+2ab62(abcd)6=3c2+2d2a26b2=3c2+2d2+2cd6a2+6b2+2ab6=3c2+2d2+2cd62(abcd)6=3c2+2d2a26b2

Since a,b,c,dQa,b,c,dQ, this implies

abcd3c2+2d2a26b2=0=0abcd=03c2+2d2a26b2=0

which is the same as

ab3c2+2d2=cd=a2+6b2ab=cd3c2+2d2=a2+6b2

Suppose that b0b0, c0c0. (I\'ll leave the solution of these cases to the reader.) Then we can rewrite the first equation as ac=db=xac=db=x, where xQxQ. Now the second equation becomes

3c2+2x2b2x2(2b2c2)(x23)(2b2c2)=x2c2+6b2=3(2b2c2)=03c2+2x2b2=x2c2+6b2x2(2b2c2)=3(2b2c2)(x23)(2b2c2)=0

This implies that x2=3x2=3 or 2b2=c22b2=c2. None of them has non-zero solutions in rational numbers.