# Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich

By A. I. Kostrikin, I. R. Shafarevich

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Extra resources for Algebra II - Noncommunicative Rings, Identities

Example text

Then p divides either a or b. Proof Since the only positive divisors of p are 1 and p, it follows that the greatest common divisor of p and a is either p, in which case p divides a, or 1. So, if p does not divide a, the greatest common divisor of a and p must be 1. 6(i) (with p, a, b in place of a, b, c). 1, but probably you were already aware that primes have that property ( just through experience with numbers). But how to prove it? The concept of greatest common divisor is the key to the short and simple proof above.

If one wishes to continue this list beyond the ﬁrst few primes, then it is not very efﬁcient to check each number in turn for primality (the property of being prime). A fairly efﬁcient, and very old, method for generating the list of primes is the Sieve of Eratosthenes, described below. 194 bc) is probably more widely known for his estimate of the size of the earth: he obtained a circumference of 250 000 stades (believed to be about 25 000 miles); the actual value varies between 24 860 and 24 902 miles.

N−k)! = n! 0! n n yn (and is known as a binomial = 1 and n 0 = n! = 1. n! 1 1 1 y which, the base case, n = 1, the theorem asserts that (x + y)1 = 10 x 1 + by the observation just made, is true. Now suppose that the result holds for n = k (induction hypothesis). Then, using the induction hypothesis, we have (x + y)k+1 = (x + y)(x + y)k = (x + y) k 0 k 1 xk + x k−1 y 1 + · · · + k k−1 x 1 y k−1 + k k yk . When we multiply this out, the term involving x k+1 is k 0 x k+1 = x k+1 = k+1 0 x k+1 , k+1 k+1 y k+1 .