Download e-book for kindle: Algebra Interactive by Arjeh M. Cohen

By Arjeh M. Cohen

The 1st interactive path overlaying first and moment 12 months algebra. ranging from such basic issues 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 diversifications. the pc algebra procedure hole is built-in all through, permitting the calculation and manipulation of mathematical gadgets. moreover, collections for Mathematica notebooks and Maple worksheets evaluation the algorithms provided. a number of selection routines offer clients with quick suggestions, whereas amenities for tracking scholars and a bulletin board entire this electronic path.

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They are just taken in different order, but that does not influence the result. In other words, the products are equal. But the latter product equals ∏z∈Z/nZ× (a· z) = aΦ(p) · ∏z∈Z/nZ× z. By Invertibility of Products the product ∏z∈Z/nZ× z is invertible, so, multiplying both sides of the above equation by its inverse, we find aΦ(n) = 1. This proves the theorem. 11. The set Z/15Z× contains 8 elements, one of them being 7(mod 15). eqsmod 78 , 494 , 44 , 12 , 1, 15 This in accordance with Euler’s Theorem.

Below the values of the Euler totient function are listed for all positive integers up to 20. 6: Euler totient function 55. 7 (Euler Totient). The Euler totient function satisfies the following properties. 1. Suppose that n and m are positive integers. 25) 2. 26) 3. 27) i=1 4. 29) d∈{d∈N|d|n} Proof. Part (a). Suppose that n and m are two positive integers which are coprime. If a and b are two integers congruent modulo n· m, then they are also congruent modulo n and modulo m. Moreover, if an integer a is relatively prime to n· m, then clearly a is also relatively prime to both n and m.

They turn out to be those nonzero elements which are not invertible. Hence a nonzero element in Z/nZ is either invertible or a zero divisor. 46. 22 (Zero Divisor Characterization). Let n > 1 and a ∈ Z. 1. The class a(mod n) in Z/nZ is a zero divisor if and only if gcd (a, n) > 1 and a(mod n) is nonzero. 2. The residue ring Z/nZ has no zero divisors if and only if n is prime. Proof. The second statement of the theorem is a straightforward consequence of the first. So, we only prove the first. There are two parts to the proof.

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Algebra Interactive by Arjeh M. Cohen

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