# Kan extensions in Enriched Category Theory by Eduardo J. Dubuc

By Eduardo J. Dubuc

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1-18 they are linearly independent. Therefore they form a basis, and [K(a):K] = n = dmM A finite extension is one whose degree is finite. Any simple algebraic extension is thus finite. However, the converse is not true. In this connection an extension L:K is algebraic if every element of L is algebraic over K. ,^). 2 Galois Theory: Solubility of Algebraic Equations by Radicals Group theory was invented by Galois to study the permutations of the zeros of polynomials. Thus any polynomial f(x) has a group of permutations of its zeros, now called its Galois group, whose structure is closely related to the methods required for solving the corresponding polynomial equation f(x) = 0.

Then/ilgi«"gj and filgi for some /. Choose the notation so that / = 1, and then/ilgi. Since/i and gl are irreducible and/i is not a constant, we must have/i = £lgl for some constant k\. ,kr are constants. The remaining gj (j > r) must also be constant or else the degree of the right hand side would be too large. ■ Methods for testing the irreducibility of polynomials are very difficult just like methods for testing the primality of numbers. 1-11: If / is a polynomial over the integers Z which is irreducible over Z, then/, considered as a polynomial over the rationals Q, is also irreducible over Q so that factorization over the integers Z is equivalent to factorization over the rationals Q.

Thus either j = / orj = -/. Now a(x + iy) - a(x) + a(i)a(y) =x +jy for any xy y e R. This leads to two candidates for R-automorphisms, namely a\:x + iy —>x + iy (the identity operation) and 0C2:x + iy —>x - iy (the complex conjugation operation). The maps ai and ot2 can be shown to be R-automorphisms. Since OC22 = a i , the Galois group T(C:R) is the cyclic group of order 2 (C2). A group M* = T(L:M) of all M-automorphisms of L can be associated with each intermediate field M. Using this terminology, KA is the entire Galois group, and L* = 1, namely the identity map on L.