## Algebra 3: algorithms in algebra [Lecture notes] by Hans Sterk

By Hans Sterk

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Additional info for Algebra 3: algorithms in algebra [Lecture notes]

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If F is a finite field only the latter case can occur and F is said to have characteristic p. 5 shows that a finite field contains a copy of the field Z/pZ with p elements. In particular, the field F is a finite dimensional vector space over Z/pZ. If the dimension is m then F has pm elements. So the number of elements of a finite field is necessarily a prime power. 1 Theorem. Let F be a field with pm elements where p is a prime. m a) Every element a ∈ F satisfies ap = a. b) The group of units F ∗ is cyclic of order pm − 1.

This implies that the solution is 1 · log(v) with v = gcd(p(θ) − q(θ) , q(θ)) = θ. So 1 = log(log(x)). 9 Example. The theorem does not apply to the integral log(x2 + 1) since the degree of the numerator (the numerator is θ = log(x2 + 1) of degree 1) is greater than the degree of the denominator. This example represents another extreme of our problem, namely where the expression is polynomial in θ. 10 Similar to the case of a single logarithmic extension is the case of a single exponential extension.

We note that there is an obvious extension of symbolic integration to differential equations, which is often called differential Galois theory. In the sequel we will work in suitable field extensions. Here, suitable means that the field is relevant to our specific class of functions, but is also adapted to the process of differentiation. The formal notion is the following. 2 Definition. A differential field consists of a field K of characteristic 0 and a map D : K → K satisfying the rules a) D(f + g) = Df + Dg for all f, g ∈ K; b) (Leibniz’ rule) D(f · g) = f Dg + g Df for all f, g ∈ K.