.is √7 ∈ Q((5+√7)^(1/2)
This gives me that the dimension of Q((5+√7)^(1/2) as a vector space over Q(√7 ) is 2.
Stop here! This already tells you that , otherwise you'd
get that ...
But i don't know if the irreducible polynomial of degree 4 over Q((5+√7)^(1/2), is still irreducible over Q(√7 ).Is it sufficient to state that a basis is [1,5+√7)^(1/2)] and
√7 cannot be written as a linear combination of this basis?