By Martin A Moskowitz
Although common in nature, this booklet offers with basic matters in arithmetic вЂ” quantity, algebra, geometry (both Euclidean and non-Euclidean) and topology. those topics, on a complicated point, are an analogous ones with which a lot of present mathematical study is worried and have been themselves study themes of past classes. the cloth is particularly compatible either for complex highschool scholars and for college kids drawn to common arithmetic from the next viewpoint. it's going to even be very helpful to school lecturers looking an summary in their material.
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This contradiction proves the corollary. • In fact this result and its proof can easily be generalized as follows below and is left to the reader as an exercise. 7. 19. Unless k 6 Z + is a perfect nth power, the nth root of k is irrational. Conversely, of course, if n € Z + is a perfect nth power, then the nth root of k is rational. 6 The Complex Numbers Notice that it follows from our axioms for an ordered field that for any x € R, x2 > 0. To see this, observe that it is clearly true if x > 0, while if x < 0 all is well since (—x)2 — x2.
1. Draw a picture of the nth roots of unity. These provide the solution to the very special case zn — 1 = 0 of the Fundamental Theorem of Algebra. 2. Use this picture to show that the sum of the nth roots of unity is always zero. 2. An nth root of unity z is called primitive, if the power zm is never 1 for m = 1 , . . , n — 1. 3. 1. Show if n is prime, then every nth root of 1 is primitive. 2. a£p(z — a). Show p is a polynomial with integer coefficients. 3. Let n > 1, and p(z) be as above. Then p(z) divides ja^j, where d is a divisor of n in Z.
Find which elements in S have norm 1 and which elements in 9 actually have inverses in g. 4. Use the same method in the ring C[x] or R[x] . Consequently, we can conclude (exercise): If a prime p € 9 divides ab, then p divides a or b (or both). Moreover, any a € 9 can be factored uniquely into primes of S. Finally, we apply some of these facts about the Gaussian integers to prove a result concerning the integers themselves. This will tell us which prime integers are a sum of squares of two integers and eventually will lead us to a famous theorem of Fermat.