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Integral Domain And Field Pdf Download

 
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MessagePosté le: Sam 3 Sep - 21:32 (2016)    Sujet du message: Integral Domain And Field Pdf Download Répondre en citant




Integral Domain And Field Pdf Download > urlin.us/3zkmq









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For the field of order 4 {0, 1, x , 1 + x } above, under addition each element has order 2 and so the additive group is the Klein 4-group (isomorphic to Z2 Z2). The multiplicative group {1, x , 1 + x } is a cyclic group of order 3 (generated by x since x2 = 1 + x and x3 = x(1 + x) = x + x2 = x + 1 + x = 1) In general the additive group of a finite field F of order pk is a direct sum of k copies of Zp , while the multiplicative group F - {0} is a cyclic group of order pk - 1.. Consider the set of things of the form {a + bx a, b Z3} with arithmetic modulo 3 and the "rule" x2 = -1 (so its a bit like multiplying in C !). Consider a, a2, a3, . Examples The rings Q, R, C are fields. This gives a field with 8 elements: {0, 1, x , 1 + x , x 2 , 1 + x 2 , x + x 2 , 1 + x + x 2 }. Theorem Every finite integral domain is a field.

Since x3 + x = 1 we have x(x2 + 1) = 1 and x-1 = 1 + x2. In the ring Z6 we have 2.3 = 0 and so 2 and 3 are zero-divisors. ) is an abelian group The distributive law. (Proof?) Definition A field is a commutative ring with identity (1 0) in which every non-zero element has a multiplicative inverse. (This explains the name.) The polynomial rings Z[x] and R[x] are integral domains. The axioms of a field F can be summarised as: (F, +) is an abelian group (F - {0}, . Exercise: Experiment by multiplying together elements to find multiplicative inverses. (Look at the degree of a polynomial to see how to prove this.) The ring {a + b2 a, b Z} is an integral domain. Proof The only thing we need to show is that a typical element a 0 has a multiplicative inverse.

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MessagePosté le: Sam 3 Sep - 21:32 (2016)    Sujet du message: Publicité

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