Abstract Algebra and Famous Impossibilities by Arthur Jones

By Arthur Jones

The recognized difficulties of squaring the circle, doubling the dice, and trisecting the perspective have captured the mind's eye of either specialist and novice mathematician for over thousand years. those difficulties, notwithstanding, haven't yielded to only geometrical equipment. It used to be simply the improvement of summary algebra within the 19th century which enabled mathematicians to reach on the miraculous end that those structures usually are not attainable. this article goals to improve the summary algebra.

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Since 0, some coefficient must be nonzero and (because there are only finitely many coefficients aj) there must be a largest i such that aj i= O. Let n be the largest such i. Hence Proof. f(X) i= f(X) = ao + a1X + a2X2 + ... + anX n where an i= O. We choose g(X) = ~ f(X). an Thus g(X) is monic, has 0' a') a zero and the same degree n as f(X). All the coefficients of g(X) are in F, moreover, since IF is a field. 2 1. If 0' is a zero of 3X3 - 2X + 1, find a monic polynomial with coefficients in Q having 0' as a zero.

Fields of the form IF (0') are essential to our analysis of the lengths of those line segments wliicu can be constructed with straightedge and compass. 1 Famous Impossibilities An Illustration: Q( J2) As Q is a subfield of C, we can consider C as a vector space over Q, taking the elements of C as the vectors and the elements of Q as the scalars. 1 Definition. The set Q(J2) ~ C is defined by putting Q(J2) = {a + bJ2: a,b E Q}. - Thus Q( J2) is the linear span of the set of vectors {I, J2} over Q and is therefore a vector subspace of Cover Q.

3 1. (a) Write down the irreducible polynomial of y'5 over Q and then prove your answer is correct. (b) Write down deg( y'5, Q) . 2. In each case write down a nonzero polynomial f(X) satisfying the stated conditions: (a) f(X) is a monic polynomial over Q with (b) f(X) is a polynomial over Q with momc. J2 J2 as a zero. as a zero but is not (c) f(X) is a polynomial of least degr ee such that f(X) E R[X] and f( V2) = O. (d) f(X) is another polynomial satisfying conditions (c). (e) f(X) E Q[X] and has both J2 and v'3 as zeros .

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