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Abstract Algebra | Ring (Mathematics) | Field (Mathematics)

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Algebra Math Notes ã Study Guide Abstract Algebra Table of Contents Rings ................................................................................................................................................... 3 Rings ............................................................................................................................................... 3 Subrings and Ideals .......................................................................................................
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   Algebra Math Notes ã Study Guide    Abstract Algebra Table of Contents Rings ................................................................................................................................................... 3   Rings ............................................................................................................................................... 3   Subrings and Ideals ......................................................................................................................... 4   Homomorphisms .............................................................................................................................. 4   Fraction Fields ................................................................................................................................. 5   Integers ............................................................................................................................................ 5   Factoring ............................................................................................................................................. 7   Factoring .......................................................................................................................................... 7   UFDs, PIDs, and Euclidean Domains .............................................................................................. 7    Algebraic Integers ............................................................................................................................ 8   Quadratic Rings ............................................................................................................................... 8   Gaussian Integers ............................................................................................................................ 9   Factoring Ideals ............................................................................................................................... 9   Ideal Classes ................................................................................................................................. 10   Real Quadratic Rings ..................................................................................................................... 11   Polynomials ....................................................................................................................................... 12   Polynomials ................................................................................................................................... 12   ℞ [   ] ............................................................................................................................................... 12   Irreducible Polynomials .................................................................................................................. 13   Cyclotomic Polynomials ................................................................................................................. 14   Varieties ......................................................................................................................................... 14   Nullstellensatz ................................................................................................................................ 15   Fields................................................................................................................................................. 17   Fundamental Theorem of Algebra .................................................................................................. 17    Algebraic Elements ........................................................................................................................ 17   Degree of a Field Extension ........................................................................................................... 17   Finite Fields ................................................................................................................................... 19   Modules ............................................................................................................................................. 21   Modules ......................................................................................................................................... 21   Structure Theorem ......................................................................................................................... 22   Noetherian and Artinian Rings ....................................................................................................... 23    Application 1: Abelian Groups ........................................................................................................ 23    Application 2: Linear Operators ...................................................................................................... 24   Polynomial Rings in Several Variables ........................................................................................... 25   Tensor Products ............................................................................................................................ 25   Galois Theory .................................................................................................................................... 27   Symmetric Polynomials .................................................................................................................. 27   Discriminant ................................................................................................................................... 27   Galois Group .................................................................................................................................. 27   Fixed Fields ................................................................................................................................... 28   Galois Extensions and Splitting Fields ........................................................................................... 28   Fundamental Theorem ................................................................................................................... 29   Roots of Unity ................................................................................................................................ 29   Cubic Equations ............................................................................................................................. 30   Quartic Equations .......................................................................................................................... 30   Quintic Equations and the Impossibility Theorem ........................................................................... 31    Transcendence Theory .................................................................................................................. 32    Algebras ............................................................................................................................................ 33   Division Algebras ........................................................................................................................... 33   References ........................................................................................................................................ 34    1   Rings 1-1   Rings  A ring    is a set with the operations of  addition (+) and multiplication ( × ) satisfying:1. R is an abelian group  + under addition.2. Multiplication is associative. In a commutative ring, multiplication is commutative. Aring with identity has a multiplicative identity 1.3. Distributive law:  +  =  +  ,  +  =  +    A left/ right zero divisor is an element ∈ so that there exists ≠ 0 with  = 0,  = 0 ,respectively. A commutative ring without zero divisors is an integral domain . A ring is a division ring if  − {0} is a multiplicative group under multiplication (inverses exist). Adivision ring is a field if the multiplicative group is abelian. Ex. of Rings:    ℞ ,  [  ] (the ring of polynomials in x, where R is a ring),  [  1 , … ,   ]  Note: A field is a UFD, though not a very interesting one.Basic properties:1.  0 = 0  = 0  2. − = − = − (  )  3. The multiplicative identity 1 is unique (if it exists).4. A multiplicative inverse is unique if it exists.5. If R is not the zero ring {0}, then 1 ≠ 0 .Group G Abelian groupRing RCommutativeringRing withidentityIntegral domainDivision ringField FPrincipal idealdomain (PID)Euclidean domainUniquefactorizationdomain (UFD)  6. If R is an integral domain, the cancellation law holds. Conversely, if the left (right)cancellation law holds, then R has no left (right) zero divisors.7. A field is a division ring, and any finite integral domain is a field.From here on rings are assumed to be commutative with identity unless otherwise specified. A unit is an element with a multiplicative inverse.The characteristic of a ring is the smallest  > 0 so that 1 + ⋯ + 1   = 0 . If this is never true,the characteristic is 0. 1-2 Subrings and Ideals  A subring is a subset of a ring that is closed under addition and multiplication. An ideal I is a subset of a ring satisfying:1. I is a subgroup of   + .2. If  ∈ , ∈ then ∈ .Every linear combination of elements   ∈ with coefficients   ∈ is in I.The principal ideal    =  generated by  is the ideal of multiples of   .The smallest ideal generated by  1 , … ,   is  1 ,  2 , … ,    =  1  1 + ⋯ +       ∈  Ideals and fields:1. The only ideals of a field are the zero ideal and the unit ideal.2. A ring with exactly two ideals is a field.3. Every homomorphism from a field to a nonzero ring is injective. A principal ideal domain (PID) is an integral domain where every ideal is principal.Ex. ℞ is a PID.If F is a field,  [  ] (polynomials in x with coefficients in F) is a PID: all polynomials in theideal are a multiple of the unique monic polynomial of lowest degree in the ideal. A maximal ideal of R is an ideal strictly contained in R that is not contained in any other ideal. 1-3 Homomorphisms  A ring homomorphism    : →′ is compatible with addition and multiplication:1.  +  =  +  (  )  2.  =  (  )  3.  1  = 1   An isomorphism is a bijective homomorphism and an automorphism is an isomorphismfrom R to itself. The kernel is { ∈ |  = 0} , and it is an ideal. 1-4 Quotient and Product Rings (11.4,6) If I is an ideal  /  is the quotient ring . The canonical map  : → /  sending → +  isa ring homomorphism that sends each element to its residue. If   = (  1 , … ,   ) , the quotient ring is obtained by “killing” the   , i.e. imposing the relations  1 , … ,   = 0 .
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