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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2017 • Question 10

Question

Math 6310 • Algebra Spring 2017 • Question 10

Let \(a,b,c,d\) integers such that \[ \det (A)\ne 0\qquad \text {where}\qquad A=\begin {pmatrix} a & b\\ c & d \end {pmatrix}. \] Let \(x\) and \(y\) be indeterminates, and set \(L=\mathbb {C}(x,y)\) and \(K=\mathbb {C}(x^a y^b,x^c y^d)\).

  1. Prove that \(L\) is a finite extension of \(K\) and that \((L:K)=|\det (A)|\).
  2. Show that \(L/K\) is Galois, and compute the Galois group.

Raw LaTeX

Let \(a,b,c,d\) integers such that
\[
\det(A)\ne 0\qquad\text{where}\qquad
A=\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}.
\]
Let \(x\) and \(y\) be indeterminates, and set \(L=\mathbb{C}(x,y)\) and
\(K=\mathbb{C}(x^a y^b,x^c y^d)\).

\begin{enumerate}
\item Prove that \(L\) is a finite extension of \(K\) and that \((L:K)=|\det(A)|\).
\item Show that \(L/K\) is Galois, and compute the Galois group.
\end{enumerate}

Topic Tags

Ext Functors and Module Extensions Field Extensions Finite Group Theory Galois Theory