Notes on Algebra
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چکیده
These notes contain as little theory as possible, and most results are stated without proof. Any introductory book on algebra will contain proofs and put the results in a more general, and more beautiful framework. For example, a book by Childs [C95] covers all the required material without getting too abstract. It also points out the cryptographic applications. By integer, we mean a positive or negative integer. We denote by Z the set on integers For an integer n, we denote by ||n|| the length of n, i.e. the number of bits needed to represent it, i.e. ||n|| = log 2 n. Logarithms will always be to the base 2, so we will omit the base hereafter. We will denote by ln n the natural logarithm of n, i.e. the logarithm taken to the base e = 2.71828. .. For integers k, n, we say that k divides n (or that k is divisor of n) if n is a multiple of k. For example 5 divides 35. We write k|n when k divides n. A prime number is a positive integer p ≥ 2 whose only divisors are 1 and p. Notice that 2 is the only even prime number. When a number is not prime, it is called composite. A composite can always be written (in a unique way) as a product of primes, possibly with repetitions. E.g. There are infinitely many prime numbers (which is very easy to prove), and in fact there are quite a lot of them (which is harder to prove). Specifically, if we define π(n) to be the number of prime numbers p such that 2 ≤ p ≤ n, then π(n) is about n/ ln n. Formally Theorem 1 (Prime Numbers Theorem) lim n→∞ π(n) n/ ln n = 1 The following bounds are also known 1
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