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)?It says any permutation sigma of x_1, ..., x_n can be extended to a bijection of Q(x_1, ..., x_n) defined by    sigma f(x_1, ..., x_n) = f(sigma x_1, ..., sigma x_n).

But I don't see how this definition can be consistent. For example, let    f(a, b) = a - b
    g(a, b) = a/a + b/b = 2

    x_1 = 5
    x_2 = 3
    sigma x_1 = x_2
    sigma x_2 = x_1

Then according to the formula:    sigma f(5, 3) = sigma (5 - 3) = sigma 2 = f(3, 5) = -2

But    sigma g(5, 3) = sigma 2 = g(3, 5) = 2

Contradiction?