I am trying to solve this exercise:
Let $F=K(a,b)$, $F/K$ is an extension field and $a,b\in F$ are algebraic over $K$ with $\deg(\mathrm{Irr}(a,K))=2$ and $\deg(\mathrm{Irr}(b,K))=3$. I am asked to calculate the degree of the extension $F/K$, denoted by $|F:K|$, and to prove that $F=K(a+b)$.
I have been able to prove that $|F:K|=6$, since $|K(a):K|=2$, $|K(b):K|=3$ and $\gcd(2,3)=1$.
However, I do not know how to approach the second part of the problem. Since not much information of the elements $a$ and $b$ is given, I have been trying to prove that $|K(a+b):K|=6$, but I have not been able to do it.
Any hints as to how to approach this problem would be very helpful. Thanks in advance.