Question

The dimension of Q(√2, √3) over field Q of all rational numbers
is :
(a) 1
(b) 2
(c) 3
(d) 4​

Answer 1

Answer:

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Step-by-step explanation:

One can use the fact that Q(2–√,3–√)=(Q(2–√))(3–√) the latter of which has elements of the form a+b3–√ where a,b∈Q(2–√) since [(Q(2–√))(3–√):Q(2–√)]=2. From here we observe that a=a1+a22–√ and b=b1+b22–√ for some rationals ai and bi since [Q(2–√):Q]=2 . Together substituting back into a+b3–√ you can see that {1,2–√,3–√,6–√} is a spanning set. Since the dimension of Q(2–√,3–√) as a vector space over Q is 4 we know that the spanning set is also a basis as desired.