# Download Abstract algebra [Lecture notes] by Thomas C. Craven PDF

By Thomas C. Craven

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**Additional resources for Abstract algebra [Lecture notes]**

**Example text**

If you examine the multiplication you will find that it is the same as homework problem 26, page 65. For example, [x][x + 1] = [x2 + x] = [1]. 3. More generally, assume that n is prime. If p(x) has degree k in Zn , there are nk polynomials which are possible remainders, so Zn [x]/(p(x)) has nk elements. 2 into the language of elements of F [x]/(p(x)). It says that it makes sense to do addition and multiplication by [f (x)] + [g(x)] = [f (x) + g(x)] and [f (x)][g(x)] = [f (x)g(x)]. One can now check all the axioms of a ring to see that indeed F [x]/(p(x)) is also a ring (just as we did for Zn ).

If G and H are both finite, then |G × H| = |G||H|. Proof. Check the four axioms: closure is clear since we defined something which makes sense. Associativity comes from associativity in G and H. The identity element of G × H is (eG , eH ). The inverse of (g, h) is (g −1 , h−1 ). The order statement for finite groups is true because the Cartesian product has that number of elements. 2 of the book is a collection of easy facts, none of which are worth calling a theorem. 5 points out some easy facts that have the same proofs as in other contexts where they hold.

We know from earlier work that it is the number of integers in {1, 2, . . , n − 1} which are relatively prime to n. This number is denoted by φ(n) and called the Euler φ-function. It is commonly studied in number theory and can be computed exactly, with the value depending on the factorization of n into primes. For example, φ(p) = p − 1 for a prime p. Other interesting infinite examples come from matrices. The general linear group GL(n, R) is the set of all invertible n × n matrices under the operation of matrix multiplication.