By David R. Finston, Patrick J. Morandi
This textual content seeks to generate curiosity in summary algebra via introducing every one new constitution and subject through a real-world program. The down-to-earth presentation is obtainable to a readership with out previous wisdom of summary algebra. scholars are resulted in algebraic recommendations and questions in a usual means via their daily reviews.
- Identification numbers and modular arithmetic
- (linear) error-correcting codes, together with cyclic codes
- ruler and compass constructions
- symmetry of styles within the genuine aircraft
Abstract Algebra: constitution and Application is appropriate as a textual content for a primary direction on summary algebra whose major goal is to generate curiosity within the topic or as a supplementary textual content for extra complicated classes. the cloth paves tips on how to next classes that additional strengthen the idea of summary algebra and should attract scholars of arithmetic, arithmetic schooling, desktop technology, and engineering attracted to purposes of algebraic concepts.
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Extra info for Abstract Algebra: Structure and Application
B/ C . a C . a/ C . 0 C . b// C b D . a b/. 15 (Cancellation Law of Addition). Let a; b; c 2 R. If a C b D a C c, then b D c. Proof. Let a; b; c 2 R, and suppose that a C b D a C c. By adding the following string of equalities. a C c/ . a C a/ C b D . a C a/ C c 0Cb D0Cc bDc Again, notice that in each line we used properties from the deﬁnition of a ring. t u The element 0 is defined with respect to addition, but we want to know how it behaves with respect to multiplication. The distributive properties are the only properties in the definition of a ring that involve both operations.
5 The Golay Code In this section we discuss a length 24 code used by NASA in the 1970s and 1980s to transmit images of Jupiter and Saturn photographed by the Voyager spacecraft. This code, called the extended Golay code, is the set of solutions to the matrix equation H x T D 0, where H is the 12 24 matrix H D ŒI j B whose left half is the 12 12 identity matrix I and whose right half is the symmetric 12 12 matrix 0 11 B1 0 B B B0 1 B B1 1 B B1 1 B B1 0 BDB B0 0 B B0 0 B B B0 1 B B1 0 B @0 1 11 01 11 11 10 00 00 01 10 01 11 10 11 11 10 00 00 01 10 01 11 10 01 11 11 000 001 010 101 011 110 101 011 111 110 100 111 10 01 11 10 01 11 11 10 00 00 01 11 1 1 1C C C 1C C 1C C 1C C 1C C 1C C 1C C C 1C C 1C C 1A 0 which satisfies B 2 D I .
Similarly for the other two codewords in C . 6. The code C D f00000; 10000; 011000; 11100g above corrects no errors. Note that the word u D 11000 given in that example is a distance 1 from a codeword, but that codeword is not the unique closest codeword to u. To determine for which t a code corrects t errors, we relate error correction to the distance of a code. 7. u; v/ W u; v 2 C; u ¤ vg : For intuitive purposes it may be useful to think of the minimum distance as the diameter of the smallest circle containing at least two codewords.
Abstract Algebra: Structure and Application by David R. Finston, Patrick J. Morandi