GATE Algebra: Fields and Extensions – Previous Year Questions

7 GATE previous year questions on Fields and Extensions (Algebra, Mathematics) with answers and explanations, from every paper.

1. GATE 2021 MA Q55 (2 marks, Numerical answer) – F is a finite field whose nonzero multiplicative group contains a subgroup of order 17. Find the smallest possible F .
2. GATE 2022 MA Q49 (2 marks, Multiple select) – Let K⊆ℂ be all elements algebraic over ℚ. Which hold?
3. GATE 2024 MA Q29 (1 mark, Numerical answer) – Find GL₂(F₃) , the number of invertible 2×2 matrices over the three-element field (integer).
4. GATE 2024 MA Q59 (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)
5. GATE 2025 MA Q47 (2 marks, Multiple choice) – K is algebraically closed and contains a finite field F. L consists of elements of K algebraic over F. S1: L is algebraically closed. S2: L is…
6. GATE 2025 MA Q60 (2 marks, Multiple select) – E⊂F⊂K are nonalgebraic field extensions. α∈K∖F is algebraic over F. L is generated over E by coefficients of the monic minimal polynomial of α over F.…
7. GATE 2026 MA Q29 (1 mark, Numerical answer) – Find the degree over ℤ₃ of the field extension Z 3[x]/ x3-x+1 (integer).