By Niels Lauritzen

Concrete summary Algebra develops the speculation of summary algebra from numbers to Gr"obner bases, whereas takin in the entire ordinary fabric of a conventional introductory direction. additionally, there's a wealthy offer of subject matters equivalent to cryptography, factoring algorithms for integers, quadratic residues, finite fields, factoring algorithms for polynomials, and platforms of non-linear equations. a distinct function is that Gr"obner bases don't seem as an remoted instance. they're absolutely built-in as a subject matter that may be effectively taught in an undergraduate context. Lauritzen's method of instructing summary algebra relies on an in depth use of examples, purposes, and workouts. the fundamental philosophy is that inspiring, non-trivial functions and examples provide motivation and simplicity the educational of summary ideas. This ebook is outfitted on numerous years of skilled educating introductory summary algebra at Aarhus, the place the emphasis on concrete and encouraging examples has more suitable pupil functionality considerably.

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**Example text**

This congruence may or may not satisfy x ≡ ±y (mod N ). To tell whether a number n is square we factor it, n = p1m 1 · · · prm r , using some predefined factor basis P = { p1 , . . , pr } of (small) prime numbers. In this context, n is a square if and only if all the exponents m 1 , . . , m r are even. Using linear algebra there is a method of systematically finding a subset {i 1 , . . 49). Let us apply this algorithm to the numbers we get from the function S(x) = 2 x − N . Notice that x 2 ≡ S(x) (mod N ).

As you may know Fermat’s last theorem haunted mathematicians for more than 300 years before it was finally proved by A. Wiles in 1994. It is unlikely that Fermat could have forseen that his little theorem would play a crucial role in generating large prime numbers for use in the modern information age. By using congruences it is easy to see (since 5 ≡ −1 (mod 6)) that 55 ≡ 5 = 1 (mod 6). 2, 6 is not a prime number. This is of course a complicated way of proving the latter, but in fact it contains the idea for some beautiful algorithms for deciding whether a number is composite without ever trying to factor it.

If a1 = 0, a2 = 2, a3 = 4, we get X = 20 + 24 = 44. This gives X = [X ]30 = 14 as expected. 7 Euler’s theorem Let (Z/N )∗ = {X ∈ Z/N | gcd(X, N ) = 1} for N ∈ N and define the function ϕ(N ) = |(Z/N )∗ |. This function is the famous Euler ϕ-function. It counts the numbers relatively prime to and smaller than a given natural number. The beginning of the table of values looks like n ϕ(n) 0 0 1 1 2 1 3 2 4 2 5 4 6 2 7 6 8 4 9 6 10 4 11 10 12 4 13 12 If you can come up with an effective way of computing ϕ you will have broken the RSA cryptosystem.