Abstract Algebra and Famous Impossibilities by Arthur Jones

By Arthur Jones

The well-known difficulties of squaring the circle, doubling the dice, and trisecting the attitude have captured the mind's eye of either specialist and beginner mathematician for over thousand years. those difficulties, although, haven't yielded to simply geometrical tools. It was once purely the advance of summary algebra within the 19th century which enabled mathematicians to reach on the incredible end that those buildings are usually not attainable. this article goals to improve the summary algebra.

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B . Fraleigh, A First Course in Abstract Algebra, 3rd edition, Addison-Wesley, Read ing, Massachusetts, 1982. H. M. Wright , An Introduct ion to the Theory of Numbers, Clarendon, Oxford, 1960. [IN1] I. Niven, Numbers: Rational and Irrational, Random Honse, New York, 1961. [IN2] I. Niven, Irrational Numbers, Carus Mathematical Monographs, No. 11, Mathematical Association of America, 1967. W. Shapiro, Introduction to Abstra ct Algebra, McGraw-Hill, New York, 1975. M. Tropp er, Linear Algebra, Thomas Nelson, London, 1969.

Let p(X) be the element of Q[X] given by p(X) = 2X 3 + 3X 2 + 2X + 3. (a) Use the Rational Roots Test to find all possible rational zeros of p(X) . (b) Is -1 a zero of p(X)? (c) Is -~ a zero of p(X)? 2. Use the Rational Roots Test to prove that 3. Find a polynomial in Q[X] which has show that V2 + v'3 is irrational. J5 is irrational. V2 + v'3 as [Hint. ] 0' = a zero. Hence V2 + v'3 and then 4. (a) Use the Rational Roots Test to prove that for each mEN, is rational if and only if m is a perfect square.

The followin g proposi tion gives a way of describing Q( J2") as a field with a certain prop er ty. 1 , in volves proving a set inclu sion. 5 Proposition. Q()2 ) is the sma llest field containing all the n umbers in the field Q and the number )2. Proof. Let IF b e a ny field containi ng Q and )2. It is clear from what has been said ear lier in t his section t hat Q( J2) is a field which contai ns bot h Q and J2. To show it is the s m lIest J such field we sha ll prove t hat Q(J2) ~ F. To prove that Q()2 ) ~ IF we shall show t hat if x E Q( V2), then 1: E IF.

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