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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2003 • Question 5

Question

Math 6310 • Algebra Fall 2003 • Question 5

Let \(G\) be the Galois group of the polynomial \(x^3-3x^2-3\) over the field \(F\). Determine \(G\) in each of the following cases:

  1. \(F\) is the complex numbers.
  2. \(F\) is the real numbers.
  3. \(F\) is the rational numbers.
  4. \(F\) is the integers modulo 2.

Raw LaTeX

Let \(G\) be the Galois group of the polynomial \(x^3-3x^2-3\) over the field \(F\). Determine \(G\) in
each of the following cases:
\begin{enumerate}
\item[(a)] \(F\) is the complex numbers.
\item[(b)] \(F\) is the real numbers.
\item[(c)] \(F\) is the rational numbers.
\item[(d)] \(F\) is the integers modulo 2.
\end{enumerate}

Topic Tags

Galois Groups of Polynomials Galois Theory Irreducible Polynomials Polynomial Rings and Ring Homomorphism Kernels