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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2015 • Question 8

Question

Math 6310 • Algebra Spring 2015 • Question 8

Suppose \(K\subset L\subset M\) are fields with \([M:L]=2=[L:K]\). Prove that \(M=K(\alpha )\), where \(\alpha \) is a root of an irreducible polynomial in \(K[x]\) of the form \(x^4+bx^2+c\).

Raw LaTeX

Suppose \(K\subset L\subset M\) are fields with \([M:L]=2=[L:K]\). Prove that \(M=K(\alpha)\),
where \(\alpha\) is a root of an irreducible polynomial in \(K[x]\) of the form \(x^4+bx^2+c\).

Topic Tags

Galois Groups of Polynomials Irreducible Polynomials Polynomial Rings and Prime Ideals Polynomial Rings and Ring Homomorphism Kernels