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Unformatted text preview: M2P4 Rings and Fields Mathematics Imperial College London ii As lectured by Professor Alexei Skorobogatov and humbly typed by [email protected] . CONTENTS iii Contents 1 Basic Properties Of Rings 1 2 Factorizing In Integral Domains 5 3 Euclidean domains and principal ideal domains 11 4 Homomorphisms and factor rings 19 5 Field extensions 29 6 Ruler and Compass Constructions 33 7 Finite ±elds 43 iv CONTENTS 1. BASIC PROPERTIES OF RINGS 1 Chapter 1 Basic Properties Of Rings Defnition 1.1. A ring R is a set with two binary operations, + and · , satisfying: ring (1) ( R, +) is an abelian group, (2) R is closed under multiplication, and ( ab ) c = a ( bc ) for all a,b,c ∈ R , (3) a ( b + c ) = ab + ac and ( a + b ) c = ac + bc for all a,b,c ∈ R . Example 1.2 (Examples oF rings). 1. Z , Q , R , C . 2. 2 Z – even numbers. Note that 1 n∈ 2 Z . 3. Mat n ( R ) = { n × nmatrices with real entries } In general AB n = BA . A ring R is called commutative if ab = ba for all a,b ∈ R . commutative 4. Fix m , a positive integer. Consider the remainders modulo m : , 1 ,..., m − 1. Notation. Write n for the set of all integers which have the same remainder as n n when divided by m . This is the same as { n + mk  k ∈ Z } . Also, n 1 + n 2 = n 1 + n 2 , and n 1 · n 2 = n 1 n 2 . The classes , 1 ,... , m − 1 are called residues modulo m . The set b , 1 ,..., m − 1 B is denoted by Z m or by Z /m or by Z /m Z . Z /m 5. The set of polynomials in x with coe±cients in Q (or in R or C ) b a + a 1 x + ... + a n x 2  a i ∈ Q B = Q [ x ] with usual addition and multiplication. If a n n = 0 then n is the degree of the polynomial. Defnition 1.3. A subring of a ring R is a subset which is a ring under the same subring addition and multiplication. Proposition 1.4. Let S be a nonempty subset of a ring R . Then S is a subring of R if and only if, for any a,b ∈ S we have a + b ∈ S , ab ∈ S and − a ∈ S . Proof. A subring has these properties. Conversely, if S is closed under addition and taking the relevant inverse, then ( S, +) is a subgroup of ( R, +) (from group theory). S is closed under multiplication. Associativity and distributivity hold for S because they hold for R . s 2 1. BASIC PROPERTIES OF RINGS Defnition 1.5. Let d be an integer which is not a square. Defne Z [ √ m ] = { a + b √ m  a,b ∈ Z } . Z [ √ m ] Call Z [ √ − 1] = b a + b √ − 1 ,a,b ∈ Z B the ring of Gaussian integers . Gaussian integers Proposition 1.6. Z [ √ d ] is a ring. Moreover, i± m + n √ d = m ′ + n ′ √ d , then m = m ′ and n = n ′ . Proof. Clearly Z [ √ d ] ⊂ C . Consider m,n,a,b ∈ Z . Then we have: Closure under addition: ( m + n √ d ) + ( a + b √ d ) = ( m + a ) + ( n + b ) √ d . Closure under multiplication: ( m + n √ d )( a + b √ d ) = ma + nbd + ( mb + na ) √ d . Also, − ( m + n √ d ) = ( − m ) + ( − n ) √ d . Hence Z [ √ d ] ⊂ C is a subring by Proposition 1.4. Finally, i± m + n √ d = m ′ + n ′ √ d , then i± n n = n ′ we write √ d = m − m ′ n ′ − n which is not possible since d is not a square. There±ore, n = n ′ hence m = m ′ . s Proposition 1.7. For any two elements r,s o± a ring, we have (1) r 0 = 0 r = 0, (2) ( − r ) s = r ( − s ) = − ( rs )....
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This note was uploaded on 11/14/2011 for the course UNKNOWN math taught by Professor Ssa during the Spring '11 term at Middle East Technical University.
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