By Niels Lauritzen
Concrete summary Algebra develops the idea of summary algebra from numbers to Gr"obner bases, whereas takin in all of the traditional fabric of a standard introductory direction. additionally, there's a wealthy provide of themes comparable to cryptography, factoring algorithms for integers, quadratic residues, finite fields, factoring algorithms for polynomials, and platforms of non-linear equations. a different characteristic is that Gr"obner bases don't seem as an remoted instance. they're totally built-in as a subject matter that may be effectively taught in an undergraduate context. Lauritzen's method of educating summary algebra relies on an intensive use of examples, functions, and routines. the elemental philosophy is that inspiring, non-trivial purposes and examples supply motivation and simplicity the training of summary innovations. 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 better pupil functionality considerably.
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Additional info for Concrete Abstract Algebra: From Numbers to Gröbner Bases
Many mathematicians before Gauss took “unique factorization” for granted. Commenting on this Gauss wrote (, Section II) However, we did not wish to omit it (the proof of unique factorization) because many modern authors have offered up feeble arguments in place of proof or have neglected the theorem completely . . The idea behind the proof of unique factorization is quite easy. Suppose we wish to prove that 2 · 3 · 11 = 5 · 13 without multiplying. Assume that 2 · 3 · 11 = 5 · 13. Then 2 | 5 · 13.
Combining the partial relations produced a sparse matrix of 569466 rows and 524338 columns. This matrix was reduced to a dense matrix of 188614 rows and 188160 columns using structured Gaussian elimination. 13 gigabyte), took 45 hours on a 16K MasPar MP-1 massively parallel computer. The first three dependencies all turned out to be ‘unlucky’ and produced the trivial factor RSA-129. The fourth dependency produced the above factorization. We would like to thank everyone who contributed their time and effort to this project.
P − 1 are quadratic residues; the other half are quadratic nonresidues modulo p. 3 Proof. We already know that the quadratic residues are [12 ], [22 ], . . , [( p − 1)2 ]. But since x 2 ≡ ( p − x)2 (mod p), we see that the quadratic residues are given by the first ( p − 1)/2 numbers [12 ], [22 ], . . , [(( p − 1)/2)2 ]. These numbers really are different. If [i 2 ] = [ j 2 ] then i 2 ≡ j 2 (mod p) and p | i 2 − j 2 = (i + j)(i − j). Therefore p | i + j or p | i − j. This is only possible if i = j, because 0 ≤ i, j ≤ ( p − 1)/2.
Concrete Abstract Algebra: From Numbers to Gröbner Bases by Niels Lauritzen