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A brand new method of conveying summary algebra, the realm that experiences algebraic buildings, reminiscent of teams, jewelry, fields, modules, vector areas, and algebras, that's necessary to quite a few clinical disciplines reminiscent of particle physics and cryptology. It offers a good written account of the theoretical foundations; additionally comprises subject matters that can't be chanced on in other places, and likewise bargains a bankruptcy on cryptography. finish of bankruptcy difficulties aid readers with getting access to the topics. This paintings is co-published with the Heldermann Verlag, and inside of Heldermann's Sigma sequence in arithmetic.
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Extra resources for Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography
Suppose that D 1 2 . From the multiplicativity of the norm we have N. / D p D N. 1 /N. 2 /: Since each norm is a positive ordinary integer and p is a prime it follows that either N. 1 / D 1 or N. 2 / D 1. Hence either 1 or 2 is a unit. Therefore is a prime in ZŒi. Armed with this norm we can show that ZŒi is a Euclidean domain. 7. The Gaussian integers ZŒi form a Euclidean domain. Proof. That ZŒi forms a commutative ring with an identity can be veriﬁed directly and easily. ˇ/ D 0. But then either ˛ D 0 or ˇ D 0 and hence ZŒi is an integral domain.
Then I is an ideal and R=I D Z=¹0º Š Z is an integral domain. Hence ¹0º is a prime ideal. However Z is not a ﬁeld so ¹0º is not maximal. Note however that in the integers Z a proper ideal is maximal if and only if it is a prime ideal. 4 The Existence of Maximal Ideals In this section we prove that in any ring R with an identity there do exist maximal ideals. Further given an ideal I ¤ R then there exists a maximal ideal I0 such that I I0 . To prove this we need three important equivalent results from logic and set theory.
4 Principal Ideal Domains and Unique Factorization In this section we prove that every principal ideal domain (PID) is a unique factorization domain (UFD). We say that an ascending chain of ideals in R I1 I2 In becomes stationary if there exists an m such that Ir D Im for all r m. 1. Let R be an integral domain. If each ascending chain of principal ideals in R becomes stationary, then R satisﬁes property (A). Proof. Suppose that a ¤ 0 is a not a unit in R. Suppose that a is not a product of irreducible elements.