Galois Theory for Beginners

John Stillwell

• Galois Theory for Beginners
  • The American Mathematical Monthly
    • Vol. 101, No. 1, Jan., 1994

The goal of classical algebra was to

• express the roots of the general n th degree equation

\[x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 = 0\]

• in terms of the coefficients
  • $a_0, \cdots , a_{n-1}, $
    ${}$

• using a finite number of

  • operation $+, -, \times, \div$ and

  • radicals $\displaystyle \;\sqrt{\;\;} \;, \;\sqrt[3]{\;\;} \;,\; \cdots$

radical

• Adjoining an element $\alpha$ to a field $F$,

• $\alpha$ is called radical if

  • some (+) integer power $\alpha^m$ of $\alpha$

  • equals $f \in F$, in which case $\alpha$ may be

  • represented by the radical expression $\sqrt[m]{f}$

Field Exetention

• $F(\alpha_1, \alpha_2, \cdots, \alpha_k)$

• radical extenstion if

  • each $\alpha_i$ is radical

$Q(x_1, x_2, \cdots, x_n)$

• symmetric $\;$ w.r.t. $\; x_1, x_2, \cdots, x_n$

Any permutation $\sigma$ of $x_1, x_2, \cdots, x_n$

\[\sigma f(x_1, \cdots, x_n) = f(\sigma x_1, \cdots, \sigma x_n)\]

• $\sigma (f+g) = \sigma f + \sigma g$

• $\sigma (fg) = \sigma f \cdot \sigma g$

• automorphism of $Q(x_1, x_2, \cdots, x_n)$