Question

Let \( K \subseteq \mathbb{C} \) be the field extension of \( \mathbb{Q} \) obtained by adjoining all the roots of the polynomial equation \( (x^2 - 2)(x^2 - 3) = 0 \). The number of distinct fields \( F \) such that \( \mathbb{Q} \subseteq F \subseteq K \) is equal to ……….. (answer in integer).

Hint:

For field extensions, analyze the roots of the polynomial and their combinations to find intermediate fields.

Answer 1

Step 1: Roots of the polynomial. The roots of \( (x^2 - 2)(x^2 - 3) = 0 \) are \( \pm\sqrt{2}, \pm\sqrt{3} \). The splitting field \( K \) is generated by adjoining \( \sqrt{2} \) and \( \sqrt{3} \) to \( \mathbb{Q} \). 
Step 2: Subfields of \( K \). The intermediate fields are: - \( \mathbb{Q} \), - \( \mathbb{Q}(\sqrt{2}) \), - \( \mathbb{Q}(\sqrt{3}) \), - \( \mathbb{Q}(\sqrt{2}, \sqrt{3}) \), - \( \mathbb{Q}(\sqrt{6}) \). 
Step 3: Counting distinct fields. Thus, there are 5 distinct fields \( F \) such that \( \mathbb{Q} \subseteq F \subseteq K \). 
Step 4: Conclusion. The number of distinct fields is \( {5} \).