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# An Introduction to Galois Theory [Lecture notes] by Steven Dale Cutkosky

By Steven Dale Cutkosky

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Perrin and C. Reutenauer 26 738 739 740 741 742 743 744 745 746 747 748 749 750 1. P RELIMINARIES Thus, for a power series f (t) = n an tn with nonegative coefficients and radius of convergence ρ, we can denote, by the expression f (r), for 0 ≤ r ≤ ρ, indifferently the sum n an r n and the value of the function defined by f for t = r, with the property that both values are simultaneously finite or infinite. Note that this statement only holds because the an are nonnegative. Indeed, consider for example f (t) = (−1)n tn .

9 Let M be a prime monoid. 1. If M contains a 0-minimal ideal, it is unique. 2. If M contains a 0-minimal right (resp. left) ideal, then M contains a 0-minimal ideal; this ideal is the union of all 0-minimal right (resp. left) ideals of M . 3. If M both contains a 0-minimal right ideal and a 0-minimal left ideal, its 0-minimal ideal is composed of a regular D-class and zero. Proof. 1. Let I, J be two 0-minimal ideals of M . Let m ∈ I \ 0 and let n ∈ J \ 0. Since M is prime, there exist u ∈ M such that mun = 0.

This ideal is a D-class and all its H-classes are groups. 13 Permutation groups In this section we give some elementary results and definitions concerning permutation groups. Let G be a group and let H be a subgroup of G. The right cosets of H in G are the sets Hg for g ∈ G. The equality Hg = Hg′ holds if and only if gg′−1 ∈ H. Hence the right cosets of H in G are a partition of G. When G is finite, [G : H] denotes the index of H in G. This number is both equal to Card(G)/ Card(H) and to the number of right cosets of H in G.