By Arthur Jones

The recognized difficulties of squaring the circle, doubling the dice, and trisecting the attitude have captured the mind's eye of either expert and beginner mathematician for over thousand years. those difficulties, besides the fact that, haven't yielded to simply geometrical equipment. It used to be basically the improvement of summary algebra within the 19th century which enabled mathematicians to reach on the excellent end that those structures will not be attainable. this article goals to advance the summary algebra.

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1 Definition. A polynomial f(X) = ao + a1X + .. + anX" in F[X] is said to be monic if its leading coefficient an is 1. Thus, for example, X2 - 2 is a monic polynomial whereas 3X2 - 6 is not. Note that both these polynomials have V2 as a zero and, for our purposes, the polynomial X 2 - 2 is somewhat nicer than 3X 2 - 6. The following proposition shows that, in the definition of "algebraic over a field", we can always assume the polynomial is monic. 2 Proposition. If a complex number a is a zero of a nonzero polynomial f(X) E IF[X] then 0' is a zero of a monic polynomial g(X) E F[X] with degg(X) = degf(X) .

Fl + J3) : Q ( J6)J. ,fl + 8. )3) over Q. 9. )3). (a) Use the fact that any m + 1 vectors in a vect or space of dimension m arc linearl y dep end en t , and your ans wer to Exer cise 8(i), t o show that {I , (3, (32, (33,(34} is linearly dependent over Q.

4 Towers of Fields The aim of this section is to produce a theorem which is of vital importance in questions involving a tower consisting of three distinct subfields IE, IF and K of C. Implicit in this set-up are three different vector spaces and the theorem to be proved will give a precise relationship between their dimensions. The three vector spaces arising from the tower are as follows. Since IE is a subfield of IF, we may take IF as the vectors and E as the scalars to give the vector space (i) IF over IE.