**Permutation Group Approach To Galois Theory**

Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say *A* and *B*, that *A*2 + 5*B*3 = 7. The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that *any* algebraic equation satisfied by the roots is *still satisfied* after the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers. (One might instead specify a certain field in which the coefficients should lie but, for the simple examples below, we will restrict ourselves to the field of rational numbers.)

These permutations together form a permutation group, also called the Galois group of the polynomial (over the rational numbers). To illustrate this point, consider the following examples:

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