Question

Find the degree for the given field extension \(Q\left(\operatorname{sqrt}(2)^{\star} \operatorname{sqrt}(3)\right.\) ) over \(Q\). Which of the following is the right choice? Explain your answer.

(A) 0 (B) 4 (C) 2 (D) 6

Answer 1

(B) 4
The field extension in question is \(Q(\sqrt{2}, \sqrt{3})\) over \(Q\). To find the degree of the extension, we consider the minimal polynomial of the elements being adjoined.
Both \(\sqrt{2}\) and \(\sqrt{3}\) are algebraic numbers, and their minimal polynomials over \(Q\) are \(x^2-2\) and \(x^2-3\), respectively. These polynomials are irreducible over \(Q\), so the degrees of the extensions \(Q(\sqrt{2})\) and \(Q(\sqrt{3})\) over \(Q\) are both 2.
Now, consider the extension \(Q(\sqrt{2})(\sqrt{3})\). Since \(\sqrt{3}\) is not an element of \(Q(\sqrt{2})\), the degree of the extension \(Q(\sqrt{2}, \sqrt{3})\) over \(Q(\sqrt{2})\) is 2 .

By the tower law, the degree of the extension \(Q(\sqrt{2}, \sqrt{3})\) over \(Q\) is the product of the degrees of the extensions \(Q(\sqrt{2})\) over \(Q\) and \(Q(\sqrt{2}, \sqrt{3})\) over \(Q(\sqrt{2})\). Hence, the degree is \(2 \times 2=4\)