GATE 2024 MA – Question 59

Algebra · Fields and Extensions · 2 marks · Numerical answer

K is the splitting field over ℚ of (X²−2)(X²−3). How many fields F satisfy ℚ⊆F⊆K? (integer)

Show answer and explanation

Correct answer: 5 to 5

Explanation

K=ℚ(√2,√3) has degree four and Galois group C₂×C₂. Its five subgroups correspond to five fields: ℚ, ℚ(√2), ℚ(√3), ℚ(√6), K. **5**.

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