By John Swallow

ISBN-10: 0511229321

ISBN-13: 9780511229329

Combining a concrete standpoint with an exploration-based method, this research develops Galois idea at a completely undergraduate point.

The textual content grounds the presentation within the inspiration of algebraic numbers with advanced approximations and basically calls for wisdom of a primary path in summary algebra. It introduces instruments for hands-on experimentation with finite extensions of the rational numbers for readers with Maple or Mathematica.

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15 Simplify into reduced form the expression (1 + 2α + 3α 2 )/(1 + α)15 , where mα,Q = X 3 + X + 1. 16 Find quotients of polynomial rings that are isomorphic to Q(α) for each of the following: (1) α = √ 2; √ (2) α = − 2; √ 3 (3) α = 3; (4) α is a root of X 2 + X + 1. 12 is a homomorphism. 18 Prove that for any fields F ⊂ K , [K : F ] = 1 if and only if K = F . B. 19 Let K be a subfield of C. Prove that K [α] is isomorphic to K [β] as vector spaces over K if and only if deg K (α) = deg K (β). 20* Prove that each number β ∈ C represented by an arithmetic combination in α over K without quotients is also represented by an arithmetic combination in α over K in polynomial form: β = n i i=0 ci α , n ≥ 0, ci ∈ K .

4 and the second in section 14. 2. If an Algebraic Number Generates a Field, So Do Its Affine Translations The preceding theorem gives a criterion for algebraic numbers other than α to generate K (α) over K , but the result does not indicate how to find such a β. We show, however, that affine translations of α – that is, elements aα + b for a, b ∈ Q – give other generators of the field, and that these alternate generators are themselves algebraic numbers. 4. Let K be a subfield of C, and let α be an algebraic number.

23 computationally explicit by showing that a system of linear equations may be constructed, the solution to which provides the coefficients of the inverse in the basis 1, α, α 2 , . . , α deg K (α)−1 , as follows: (a) express β as linear combination of the basis elements, as in β = deg K (α)−1 i=0 bi α i ; (b) let an arbitrary element of K [α] be written in the same basis, with indeterminate coefficients gi : γ = deg K (α)−1 i=0 gi α i ; (c) show that finding the inverse of β is equivalent to finding the gi such that deg K (α)−1 i=0 gi α i deg K (α)−1 bi α i = 1; i=0 (d) show further that the product on the left-hand side can be expressed in reduced form modulo mα,K ; (e) show then that in the reduced form, the coefficients of the α i are linear expressions in the variables gi with coefficients coming from K ; (f ) note that for the two sides of the equation to be equal, coefficients of like powers of α must be equal, so that we have a system of linear equations that may be solved to find the inverse of β; (g) show that Gaussian elimination, or some other algorithm of linear algebra, finishes a constructive method of solution.

### Exploratory Galois Theory by John Swallow

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