Galois Theory for Beginners (2010) [pdf]
Can anyone help me understand what is happening at the bottom of page 23 (page 3 of the PDF)?<p>It says any permutation sigma of x_1, ..., x_n can be extended to a bijection of Q(x_1, ..., x_n) defined by<p><pre><code>    sigma f(x_1, ..., x_n) = f(sigma x_1, ..., sigma x_n).
</code></pre>
But I don&#x27;t see how this definition can be consistent. For example, let<p><pre><code>    f(a, b) = a - b
    g(a, b) = a&#x2F;a + b&#x2F;b = 2

    x_1 = 5
    x_2 = 3
    sigma x_1 = x_2
    sigma x_2 = x_1
</code></pre>
Then according to the formula:<p><pre><code>    sigma f(5, 3) = sigma (5 - 3) = sigma 2 = f(3, 5) = -2
</code></pre>
But<p><pre><code>    sigma g(5, 3) = sigma 2 = g(3, 5) = 2
</code></pre>
Contradiction?