By Arjeh M. Cohen

The 1st interactive path protecting first and moment yr algebra. ranging from such primary themes as integers and divisions, modular mathematics and polynomials the content material extends to jewelry, fields and permutation teams. The hypertext is written in Java-enhanced HTML, and Java applets illustrate the speculation whereas additionally contributing interactive calculators for computing with integers, polynomials and variations. the pc algebra approach hole is built-in all through, permitting the calculation and manipulation of mathematical gadgets. moreover, collections for Mathematica notebooks and Maple worksheets assessment the algorithms awarded. a number of selection routines supply clients with fast suggestions, whereas amenities for tracking scholars and a bulletin board whole this electronic path.

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**Example text**

2. 10) Observe that there are exactly g distinct solutions. 3. In order to find all solutions to the congruence 24· x ≡ 12 (mod 15) we first compute the gcd of 24 and 15. 12) Instead of using the algorithm, we can also use the expression of the gcd as a linear combination of 24 and 15 to argue what the solution is. To this end, multiply both sides of the equality 3 = 2· 24 − 3· 15 by 4. This gives 12 = 8· 24 − 12· 15. So, a solution of the congruence is x = 8(mod 15). Other solutions can be found by adding multiples of 15/3(mod 15) to this particular solution.

The invertible elements in Z/2n Z are the classes x(mod 2n ) for which x is an odd integer. Indeed, the gcd of x and 2n equals 1 if and only if x is odd. An arithmetical system such as Z/pZ with p prime, in which every element not equal to 0 has a multiplicative inverse, is called a field, just like Q, R, and C. Suppose that n and a are integers with n > 1 and gcd (a, n) = 1. The Characterization of Modular Invertibility not only gives the existence of the inverse of a(mod n) in Z/nZ, but also a way to compute this inverse.

A neutral element for the multiplication is 1(mod n), as a(mod n)· 1(mod n) = a(mod n) and 1(mod n)· a(mod n) = a(mod n). The set Z/nZ together with addition and multiplication is an example of a quotient ring, an algebraic structure to be discussed in the theory of rings and fields. In Z/nZ we can add, multiply, and subtract. But how about division? Does every nonzero element have an inverse? 13. An element a ∈ Z/nZ is called invertible if there is an element b, called inverse of a, such that a· b = 1.

### Algebra Interactive by Arjeh M. Cohen

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