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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1995 • Question 4

Question

Math 6310 • Algebra Fall 1995 • Question 4

Suppose \(F\) is a field of characteristic 0 with the property that every polynomial of odd degree over \(F\) has at least one root in \(F\). Assume that \(E\) is an extension of degree 2 over \(F\) with the property that for each \(\alpha \in E\), the polynomial \(x^2-\alpha \) has a root in \(E\). Prove that \(E\) is algebraically closed.

Raw LaTeX

Suppose \(F\) is a field of characteristic 0 with the property that every polynomial of odd degree
over \(F\) has at least one root in \(F\). Assume that \(E\) is an extension of degree 2 over \(F\) with
the property that for each \(\alpha\in E\), the polynomial \(x^2-\alpha\) has a root in \(E\). Prove
that \(E\) is algebraically closed.

Topic Tags

Ext Functors and Module Extensions Field Extensions Galois Groups of Polynomials Polynomial Rings and Ring Homomorphism Kernels