GATE 2022 MA – Question 49
Let K⊆ℂ be all elements algebraic over ℚ. Which hold?
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Correct answer: (A) No element of ℂ∖K is algebraic over ℚ.; (B) K is an algebraically closed field.; (C) Every bijective ring homomorphism f:ℂ→ℂ has f(K)=K.
Explanation
A is the definition. Algebraic numbers form a field. A root of a polynomial with algebraic coefficients is algebraic over their finite extension of ℚ, and thus over ℚ; all polynomial roots belong to K, proving B. A complex-field automorphism fixes ℚ and preserves polynomial equations, and its inverse does likewise, proving C. K is a countable union of finite root sets of rational polynomials, so it is countably infinite like ℚ and D is false. **A,B,C**.