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# Modern Algebra MATH 6303

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ABSTRACT ALGEBRA MODULUS SPRING 2006 by J utta Hausen University of Houston Undergraduate abstract algebra is usually focused on three topics Group Theory Ring Theory and Field Theory Of the myriad of text books on the subject the following references will be used D John R Durbin Modern Algebra Fourth Edition Wiley amp Sons New York NY 2000 ISBN 0 471 32147 8 GG J Gilbert and L Gilbert Elements of Modern Algebra Fifth Edi tion BrooksCole Paci c Grove CA 2000 ISBN 0 534 37351 8 R J J Rotman A First Course in Abstract Algebra Second Edition Prentice Hall Upper Saddle River NJ 2000 ISBN 0 13 011584 3 These notes are intended for mathematics students as a compact summary of undergraduate abstract algebra 1 FUNDAMENTALS The symbols NZQR C denote the set of all positive integers all in tegers all rational numbers all real numbers and all complex numbers respectively You are familiar with mathematical induction with the fact that N is a welliordered set and concepts from elementary number theory like primes greatest common divisors and the division algorithm Exercise 1 Prove If a b and c are integers such that a divides be and ii a and b are relatively prime then a divides c Hint Try to mimic the proof of Euclid s Lemma R page 43 You are also familiar with elementary set theory intersection and union of sets subsets and set builder notation like n n is a positive integer which of course equals N The cardinality number of elements of a set X is denoted by You are able to prove equalities like A O B U C A O B U A O C for all sets A B C GG page 8 Example 13 Exercise 2 Prove A U B O C A U B O A U C for all sets A B C You also are familiar with functions also called mappings or maps and the concepts of a function f A a B being oneitoione injective or onto surjective or a oneetoione correspondence a bijection D page 11713 Given functions f A a B and g B a C the composition gof A a C is de ned by g o fa gfa for all a E A The composition of two onto functions when de ned is another onto function D page 16 21 Exercise 3 Prove If f A gt B and g B gt C are two functions which are both oneitoione then the composition 9 o f A gt C is oneitoione A function f A a B is invertible if and only if it is both oneitoione and onto If f is invertible there exists a unique function f 1 B a A such 1 2 that f o f 1 13 and f 1 o f 1A 1X denotes the identity function on the set X Exercise 4 Proue ff A a B is a function and if there exists a function g B a A such that f 09 13 andgo f 1A then f is inuertible and 9 1 An important relation on the set Z of all integers is congruence modulo m where m 2 0 denotes some xed integer that is called the modulus De ne two integers a and b to be congruent modulo m if m divides a E I Write a E 1 mod m ifa and b are congruent modulo m Notice that a E 1 mod 0 if and only if a I while a E 1 mod 1 for any two integers a and 1 Thus in most cases the modulus m is assumed to be 2 2 Congruence modulo m is an equivalence relation on Z ie the relation E mod m is re exive symmetric and transitive R page 63 145 The equivalence class containing a is denoted by a and called the congruence class ofa modulo m Suppose m 2 2 and let a E Z By the division algorithm there exist unique integers q and r such that a qm r and 0 g r lt m This unique r is said to be the remainder after diuiding a by m R p 37 De nition Two integers a and b are congruent modulo m if and only if they have the same remainder after division by m R page 63 146iii Each integer a is congruent modulo m to exactly one element in the set 01 m E 1 R page 64 147 It follows that the set Zm a a E Z of all congruence classes modulo m is a nite set of cardinality m in fact GG page 83 Zm 01m71 Congruence modulo m is compatible with the operations of addition and multiplication ofintegers in the sense that a E 1 mod m and a E 6 mod m imply that a a E b 6 mod m and that aa E 66 mod m R page 64 148 2 GROUPS A group is a pair G 9 where G is a set and a binary operation on G which associates with every ordered pair a b E G x G a unique element a gtk b E G such that the following conditions hold a For all abc G abca 120 b There exists e E G such that e a a e a for all a E G c For every a E G there exists a E G such that a a a a e If a b b a for all elements ab in a group G then G is said to be a commutatiue or an abelian group The order of G is the cardinality of the set G Suppose G 9 is a group One can show that there exists one and only one element e E G that has property b and this is called the identity element of G also given a E G there exists one and only one a E G satisfying c and this a is called the inuerse of a if the operation is considered to be a multiplication a is denoted by a l if is considered to be an addition 3 one writes a 7a and calls a the additive inverse of a or the negative of a Examples of Groups 2 e O7 inverse negative Q e O7 inverse negative R e 07 inverse negative C 1 e 07 inverse negative 5 Zm where 1 g m E Z and for a b E Zm a b a b e 07 and 7M m 7 a fat for all a E Zm 6 De ne Qi 1W and C to be the set of all nonzero rationals7 all nonzero reals7 and all nonzero complex numbers7 respectively Then Q7 7 1W7 and Ci7 are groups with e 1 the inverse of a is i fl 7 De ne QJr and R1 to be the set of all positive rationals and all positive reals7 respectively Then Q and R1 are groups with e 1 the inverse of a is E f1 8 17 and 1717 with 17 71 E Z 9 where denotes the set of all m x n matrices over a eld F and the operation is matrix addition e is the zero matrix of size in x n inverse of A E is 7A 10 GLn7 F7 where GLn7 F denotes the set of all invertible matrices of size nxn over the eld F and the operation is matrix multiplication e I the n x 71 identity matrix 11 SLn7 F7 where SLn7 F denotes the set of all n x nematrices over the eld F which have determinant 1 and the operation is matrix mul tiplication e I the n x 71 identity matrix 12 SX7 o where X is a nonempty set and SX is the set of all bijections B X a X with operation composition of functions e 1X7 the identity function on X 13 Smo where 5 SX with X 12n7 the group of all permutations of 17 2 7n with operation composition of functions If B 6 Sn use matrix notation for 3 Write P930 3 7 lt 1 2 n gt 61 62 301 39 Note that 5 has order nl ln general7 for 043 6 Sn 04 o B 7 B o a The identity function is a bijection7 thus 71 7 12 7i 5 MM 1 2 n 39 The inverse of B 6 5 can be found by interchanging the rows of the matrix representing 3 For example7 the inverse of 12 6lt2 413gt S4 17 2413 5 lt1234gt 471234 5 lt314239 Let G 9 be a group A subgroup of G is a subset H of G such that H 9 is a group on its own right If H is a subgroup of G this is indicated by writing H g G For example Z Q g R g C similarly 1 171 Q g R g C and SLn F GLn F when F is a eld Exercise 5 Prove For every group G e g G and G g G Exercise 6 Is QJr Q Is RJV R Is 23 Z4 Is 53 S4 Two Subgroup Criteria For a subset H Q G of a multiplicative group G the following conditions are equivalent 1 H is a subgroup of G 2 H is nonempty and ab E H implies ab 6 H and of1 E H 5 H is nonempty and ab E H implies ab 1 6 H GG page 123f 39 and 310 If G is an additive group and H Q G then H g G is equivalent to the additive versions versions of 2 and 3 ie 2 H is nonempty and ab E H implies that a b E H and 7a E H 3 H is nonempty and ab E H implies a 7 b E H which equals Integral powers and multiples Let a E G where G is a multi plicative group De ne a0 e a1 a a2 a a etc ie for n E N a is the product of n factors each of which equals a de ne a a 1 The Laws of Erponents hold For all integers in and n amn ama and am am If a E G where G is an additive group one writes in tegral multiples instead of powers Thus 0a e la a 2a a a etc ie for n E N na is the sum of n terms each of which equals a de ne ina n7a The Laws of Multiples are For all integers in and n m na ma na and mna mna Cyclic Subgroups Let G be a multiplicative group and a E G If H is a subgroup of G containing a then by the subgroup criteria H also contains a l and closure of the operation in H implies aa 1 e a0 E H closure also implies that for every positive integer n a and a 1 a must belong to H Thus ak k e Z g H It turns out that the set of all integral powers of a forms a subgroup of G called the cyclic subgroup generated by a and denoted by a This is the smallest subgroup of G containing the element a If G is an additive group we write integral multiples instead of powers In this case the cyclic group generated by a is a na n E Z For example for 2 E QJV 5 2 2 n E Z while for 2 E Q 2 n2 n E Z which is usually denoted by 2Z ls Q a group Homornorphisms Throughout G o and H 9 are groups with iden tity elements ea and eH respectively Notation will mostly be multiplica tive eg write of1 for the inverse of a group element a A homomorphism from G to H is a mapping Oz G a H such that aa o b aa ab for all a b E G Examples Each of the following maps is a homomorphism 1 Go Z and H RJV de ne Oz Z a RJV by 0471 2 for all n E Z 2 G o GL2R and H Ri de ne B GL2R a R by 3A detA for each A E GL2R 3 G o Z and H Zm where m 2 1 is some xed integer Then de ne 39y Z a Zm by 39yk for all k E Z 4 G o 24 and H Ci de ne 6 Z4 a 1 by 6k ik for all E Z4 where i 71 5 G o RJV and H R de ne 4p Rt a R by 4pz lnz for each x E R5 Exercise 7 Prove that each of the ve maps is a wellide ned homomor phz39sm Notice that the rst and the fourth maps are oneitoione but not onto the second and third maps are onto but not oneitoione and the last map is a homomorphism which is both onektoione and onto This prompts De nition An isomorphism from G to H is a homomorphism from G to H which is both oneitoione and onto Two groups G and H are isomorphic if there exists an isomorphism from G onto H If G and H are isomorphic this is symbolized by writing G E H Thus from Example 5 the groups RJV and R are isomorphic and so are the groups Z4 and the cyclic subgroup of 0 generated by i xil Exercise 8 Suppose that 04 G gt H is an isomorphism Prove that 071 H gt G is an isomorphism De nition Let Oz G a H be a homomorphism Then a The image of 04 is Im 04 h E H h 049 for some 9 E G b The kernel ofa is the set Ker Oz g E G 049 eH Exercise 9 Find the image and the kernel of each of the ve homomor phisms in the Examples above Proposition Let Oz G a H be a homomorphism Then 1 aeG 5H 2 For allg E G ag 1 ag 1 5 Ima is a subgroup of H 4 Kera is a subgroup of G 5 04 is oneitoione if and only if Kera ea 6 For all h E Kera and for allm E G m o h o x71 E Kera Exercise 10 Proue this proposition De nition A normal subgroup of a group G o is a subgroup N of G such that zonoz l ENfor allz E Gand for alln EN Thus the last part of the Proposition above may be restated by saying that the kernel of a group homomorphism is always a normal subgroup of the domain group Note that every subgroup of an abelian group is normal Also for any group G the triuial subgroup egg and the group G itself are normal subgroups of G Cosets Let K be a subgroup of a group G o and let x E G The left coset ofK in G containing m is the set zoKzok hEK Notice that e0 0 K K so that the subgroup K itself is a left coset of K in G Examples 1 Let m 5 and K 52 Z For any 2 E Z the congruence class of z modulo 5 is z K the left coset of K in Z containing 2 2 Let K S2R G2R For a matrix A E S2R the left coset A S2R consists of all matrices in G2R which have the same determinant as A A partition of a nonempty set X is a collection 73 of subsets of X such that no member of 73 is empty ii any two distinct members Of 73 are disjoint and iii the union of all subsets in 73 equals X Let G o be a group with subgroup K and z E G then x z 050 E z o K proving z oK is a nonempty subset of G In fact one has the following result R page 140 Lemma 231 Proposition Let K be a subgroup of the group Go Then the set 73 m o K m E G of all left cosets ofK in G forms a partition of G Exercise 11 Suppose that G o is a nite group K is a subgroup of G and m E G Proue that lonl Lagrange7s Theorem Let K be a subgroup of the nite group G o and let denote the set of all left cosets ofK in G Then lGl Exercise 12 Proue Lagrange s Theorem Exercise 13 Suppose G is a group of nite order 12 and K is a subgroup of G Find all integers m which might be equal the order of K Quotient Groups Let G o be a group and let N be a normal subgroup of G Consider the set of all left cosets of N in G and denote it by GN GNmole G Exercise 14 Find GN in each of the following cases a G o 53 o and N lt3 with 51 2 52 353 1 b Go Z and N 7712 where m 2 2 is some xed integer Theorem Let Go be a group and let N be a normal subgroup of G De ne an operation also denoted by 0 on the set GN by onoyoN x o y o N for all xy E G Then a This product of cosets is well de ned b GNo is a group with identity eaN ea oN N for each m E G moN 1 m l oN c The mapping ll G gt GN de ned by 11m m o N for allm E G is a surjectiue homomorphism from G to GN and Kerl N Exercise 15 Proue this theorem De nition The group GN o ofthe Theorem above is called the quotient group or factor group of G modulo N and the surjective homomorphism 1 G a GN is said to be the natural homomorphism from G to its quotient group GN The Isomorphism Theorem for Groups A homomorphic image of the group 10 is any group G with the property that there exists a homomorphism n G a G from G onto G ie G lmn Thus if 04 G a H is a homomorphism of groups then lma is a homomorphic image of G Examples From the ve examples of homomorphisms on page 5 of these notes one observes 1 2 g R1 is a homomorphic image of Z 1W is a homomorphic image of GL2R For each integer m 2 1 Zm is a homomorphic image of Z The cyclic subgroup g C is a homomorphic image of Z4 R is a homomorphic image of R1 0 pr The following fact is of fundamental importance in group theory For a proof see R page 166 Theorem 253 or D page 109 Theorem 231 The First Isomorphism Theorem Let 10 and H be groups and let oz G gt H be a homomorphism Then Kera is a normal subgroup of G and ma is a subgroup ofH which is isomorphic to the quotient group GKera Exercise 16 Consider the ue examples of homomorphisms on page 5 of these notes For each of these nd a quotient group of G which is isomorphic to the image and ii specify a mapping from this quotient group of G to the image which is an isomorphism Exercise 17 Let G be a multiplicatiue group with identity element e E G a Is G Gef2 Justify your answer b Describe the quotient group GG What is its order Exercise 18 Let G be a multiplicatiue group with identity element 5 and leta E G Proue a If a is an in nite set then a is isomorphic to the additive group Z of all integers b If a has nite order m then a is isomorphic to the additive group Zm Hint Argue that c5 k H ak k E Z is a surjective homomorphism from Z to a and that Ker gt mZ Exercise 19 Prove Being isomorphic is an equivalence relation on the collection of all groups Exercise 20 Prove If a and b are two cyclic groups of equal order then a and b are isomorphic Hint Exercise 18 above Exercise 21 Prove The multiplicative group of all nonzero real numbers is isomorphic to the quotient group GL2RSL2R 3 RINGS A ring is a triple R l where R is a set and and are two binary operations on R satisfying the following conditions a R is an abelian group with identity element 0 03 b For all a bc E R abc abc c For all abc E R abc abac and abc ac be If there exists an element 1 1R 6 R such that 1a a1 a for all a E R then R is said to be a ring with identity if ab ba for all ab E R then R is said to be a commutative ring Examples Z QRC are all commutative rings with identity the ring E of even integers is commutative but does not have an identity Given a ring R the set MAR of all n x nematrices with entries in R is a ring under the usual addition and multiplication of matrices If n 2 2 and R is a ring with identity 1 7 0 then MAR is a ring with identity namely the n x n identity matrix but MAR is not commutative The set RM of all polynomial functions in the indeterminate z with real coefficients is a ring under the usual addition and multiplication of polynomials For any integer m 2 1 Zm l is a commutative ring with identity when multiplication is de ned by 6 k 6 for all hi 6 Z Excercise 22 Prove that multiplication in Zm is well de ned If R is a ring with identity 1 one can show that 1 is unique A unit of a ring R with identity is any element u E R for which there exists v E R satisfying uv vu 1 Again one can show that given a unit u the element v with the property uv vu 1 is unique thus v is called the inverse of u and denoted by v u l Exercise 23 Let R be a ring with identity 1 Prove a1R OR if and only ifR OR 17 The set UR of all units in R is a group under the operation of mul tiplication de ned in R 9 A eld is a commutative ring F with identity 1 7 0 such that every nonzero element of F is a unit7 ie UF F 7 Examples of elds are Q7R7C and Zp when p is a prime Ring Homomorphisms Let R and S7 be rings A ring homomorphism from R to S is a mapping oz R a S such that aa b aa 041 and aab aaab for all ab 6 R Examples Each of the following maps is ring a homomorphism 1 Let R be the ring of integers and let S Zm be the ring of integers modulo m for some in 2 1 De ne oz Z a Zm by 134k for all k E Z 2 Let R R and let S M2R be the ring of all real 2 x 27matrices De ne oz R a M2R by by 04z 3 8 for all z E R 3 Let R Z and S MAC7 and de ne oz Z a MAC by 134k k k E Z7 where I denotes the n x n identity matrix 4 LetR SCand de nea CHbe aabi aibiwhere a7 1 E R Exercise 24 Prove that each of the four maps is a ring homomorphism De nition A ring isomorphism from a ring R to a ring S is a ring homo morphism oz R a S which is both oneitoione and onto Two rings R and S are isomorphic if there exists a ring isomorphism from R onto S If R and S are isomorphic7 this is symbolized by writing R E S Examples These rings are isomorphic 1 Given an nidimensional vector space V over a eld F7 the set LV7 V of all linear transformations T V a V with pointwise addition and com position of mappings as multiplication is a ring which is isomorphic to the ring of all n x nematrices over F 2 Example 4 above is an isomorphism from the eld of complex numbers to itself7 also called an automorphism Exercise 25 Prove If 04 R gt S is an isomorphism of rings then of S gt R is an isomorphism of rings De nition Let oz R a S be a ring homomorphism De ne a The image ofa is the set Im 04 y E S y 04m for some z E R b The kernel of 04 is the set Ker oz z E R 04m 05 Notice that7 if 04 R a S is a ring homomorphism7 then 04 is also homo morphism from 137 to S Thus7 kernels and images of ring homo morphisms give nothing new A subring of a ring R is a subset S of R which is a ring under the same operations as those in R Exercise 26 Let R be a ring and let S Q R Prove S is a subring ofR if and only if S is a subgroup of 137 and ii S7 is closed ie x71 6 S implies my 6 S Examples Each of these are subrings 1 The set 5Z is a subring of the ring of integers 2 The set of all upper triangular matrices in MAR is a subring of the ring of all real n x nematrices Ditto for the set of all lower triangular matrices and the set of all diagonal matrices in Proposition Let R and S be rings and let oz R gt S be a ring homomor phism Then 1 MOB 05 2 For all a E R 047a 704a 5 Ima is a subring of S 4 Kera is a subring of R 5 04 is oneitoione if and only if Kera 0R 6 For all k E Kera andfor allm E R wk 6 Kera and km E Kera Exercise 27 LetR be a ring a Prove a0R OR ORaforalla R b Prove the Proposition on ring homomorphisms stated above Ideals An ideal of a ring R is a subring I of R which is closed under externaliinternal multiplication in the sense that i E I implies that xi 6 I and im 6 I for all z E R Thus part 6 of the the proposition above implies that the kernel of a ring homomorphism is always an ideal of the domain ring Given any ring R both 03 and R are ideals of R For any xed integer n the set nZ of all integral multiples of n is an ideal of the ring of integers For example the ring lE 22 of even integers is an ideal of Z In ring theory ideals take on the role that normal subgroups play in group theory namely they allow you to de ne quotient structures Quotient Rings Let R be a ring and let I be an ideal ofR Then I is a subgroup of R which must be normal since R is a commutative group Thus the set RI aIla R of all left cosets of I in the group R is a group under the operation a I b I ab I and RI is an abelian group since R is abelian Also from group theory the mapping 1 R a RI de ned by 1a a I for all a E R is a surjective group homomorphism from R to RI Exercise 28 Let I be an ideal of the ring R and let a ab b E R such thataIaI andbIbI Prove that abI abI Theorem LetR be a ring and letI be an ideal ofR De ne a multiplication on the quotient group RI by aIbI abI for all ab E R Then a This multiplication is well de ned b RI is a ring with ORI 0R I I ifR is a ring with identity 13 then so is RI and 1131 13 I c The mapping 1 R gt RI de ned by 1a a I for all a E R is a surjective ring homomorphism from R to RI and Kerl I Exercise 29 Prove this theorem 11 De nition The ring RI of the Theorem is called the quotient Ting of R modulo I The Isomorphism Theorem for Rings Let R and S be rings A homomoiphic image ofR is any ring R with the property that there exists a ring homomorphism n R a R from R onto R Examples From the rst three examples of ring homomorphisms above one observes 1 For every integer in 2 1 the ring Zm is a homomorphic image of Z 2 The subring of M2R consisting of all real diagonal 2 x 27matrices with 2 27entry zero is a homomorphic image of the eld R 3 The set of all matrices of the form kI 6 MAC with h an integer and I the identity matrix is a subring of MAC and a homomorphic image of Z A proof of the following theoreom can be found in R page 280 Theorem 371 or GG page 251 Theorem 613 The First Isomorphism Theorem for Rings Let R and S be rings and let Oz R gt S be a ring homommphism from R to S Then Keroz is an ideal of R and Ima is a subring of S which is isommphic to the quotient ring RKeroz Exercise 30 Let R be a commutatiue ring with identity 1 7 0 and let I be an ideal of R Proue If I 7 R then RI is a commutatiue ring with identity 1 7 0 Exercise 31 Let R be a commutatiue ring with identity 1 7 0 and let a E R Proue a The set Ra pa r E R is an ideal ofR containing a Ra is called the principal ideal generated by a and is also denoted by a b IfR has no ideals other than R and 0 then R is a eld Exercise 32 Let R be a ring and let I be an ideal of R a Proue IfJ is an ideal ofR such thatI Q I then the set II jI jE J is an ideal ofRI b Suppose 7 Q RI is an ideal ofRI De ne I m E R m I E j Proue I Q I and I is an ideal of R 4 FIELDS One of the most useful applications ofthe lsomorphism Theorem for Rings occurs in the study of elds One reason is that the set of all polynomi als in an indeterminate z over a eld F is a commutative ring with identity 1 7 0 which has the property that the product of any two nonzero el ements is nonzero and ii every ideal of is principal see Exercise 31 of these notes A commutative ring with identity 1 7 0 which satis es conditions and ii is called a principal ideal domain or a PlD for short Throughout F will denote a eld For p prime 2 is a eld Many texts replace congruence classes modulo p by their unique representatives in the set 0p7 1 so that 21 0p71 Polynomials over F A polynomial over F in the indeterminate z is an expression of the form n fa0a1xanmnZaZmi l i0 where n 0 is an integer and ai E F i On If m gb0b1zmbmmmzbm 2 i0 is another polynomial over F then we agree that f 9 if and only if there exists an integer k such that ai b for all i O k and aj 0 and bi 0 for all j gt k The zero polynonial is 0 0 0m 00m 0z If f is a nonzero polynimial then one can write fa0a1manxnan7 0 3 and n is called the degree of f in symbols n degf A constant polynomial is one of the form It c with c E F Nonzero constant polynimials have degree one and the degree of the zero polynomial is unde ned If f 220 aizi and g 20 bizi are polynomials over F one de nes fg 2n maibizi and fg Ezraquot 011139 where fori O nm ci aobi albi1 aibo This de nition of course requires at 0 whentgtnandbt0whentgtm For the proof of the following proposition see GG page 294 84 page 296 85 and page 298 87 noting that elds are integral domains Proposition The polynomial ring ouer a eld F is a commutatiue ring with identity 1 1F and 0 OF The set of constant polynomials is a subring of which is isomorphic to F In fact for a E F the constant polynomial h a will identi ed with a E F If f g E are nonzero their product fg is nonzero and degfg degf degg Every polynomial 1 over F gives rise to a polynomial function from F to F namely de ne fz 220 aizi for z E F if f L0 aimi If F R it is true that two polynomials are equal if and only if they yield the same function from R to R Thus a polynomial over R is identi ed with its polynomial function and written not just as 1 but as This is done in calculus However ifF Zg 0 1 the polynomials 1z 1z21z3 all de ne the same polynomial function when evaluated on Zg But by our de nition of polynomials over a eld these are distinct The Division Algorithm Let F be a eld and let 1 E be a nonzero polynomial Then giuen any 9 6 FM there exist unique q r E such thatg 1 q r and either r 0 or ii 7 7 0 and degr lt degf For a proof see GG page 301 Also note Example 1 GG page 303 which may serve as a model for your solution of Exercise 33 Exercise 33 Let f 2x 2 and g 3 2x 2 a Find qr E R such that g fq r b Find qr E 23 such that g fq r A consequence of the Division Algorithm is the following fact See R page 245 339 for a proof Corollary IfF is a eld then is a Principal Ideal Domain PID Exercise 34 Proue that f E is a unit if and only 1 is a nonzero constant polynomial Irreducible Polynomials A polynomial p over a eld F is said to be irreducible if p has positive degree thus p is not a constant polynomial hence not zero and not a unit in and ii p 9 h with g h E implies that either 9 is a constant or h is a constant Given any nonzero constant 0 every p E has the trivial factorization p Cc lp The point is that these are the only factorizations that p admits ifp is irreducible Theorem Euery polynomial f ofpositiue degree ouerF is either irreducible or is a product of irreducible polynomials over F The proof is by induction on the degree of 1 There is even a uniqueness property which holds for this factorization but we shall not need this See D page page 166 GG page 312 824 or R page 261 352 for a proof Maximal Ideals An ideal M in a ring R is said to be a maximal ideal of R if M 7 R and ii M Q I with I an ideal of R implies M I or I R For example ifp is a prime then p 192 is a maximal ideal of Z Exercise 35 Proue IfF is a eld then 0 is a maximal ideal of F Exercise 36 Let R be a commutatiue ring with identity 1 7 0 and let M be a maximal ideal of R Proue The quotient ring RM is a eld Hint Exercises 31 and 32 of these notes Proposition Letp E be irreducible Then the principal ideal p of generated by p is maximal Proof Suppose p E is irreducible Then p 7 for otherwise 1 E p and p would be a unit contradicting degp gt 0 see Exercise 34 Let I be an ideal of containing Since is a PlD there exists 1 E such that I Now p E p Q I implies p fg for some 9 E lrreducibility implies that f is a unit or g is a unit If f is a unit then I f if g is a unit then f pg 1 6 p from which we obtain that I 1 Q p Q I and I This proves p is maximal Roots of Polynomials If f 20 aixi E and u E F de ne u 20 aiui Clearly u E F A root of f in F is any element 1 E F such that u 0 You all have heard of the question vexing mathematicians before they invented irrational numbers How could there be a root of the polynomial f x2 7 2 E QM The same struggle took place when mathematicians were disputing whether there could be a number called i for imaginary that s a root of x2 1 E The conclusion of this refresher course on undergraduate algebra will consist of an argument that for any nonconstant polynomial 1 over any eld F there exists an extension eld E of F in which 1 has a root The amount of work needed to prove this will depend on the de nition of the word extension eld Let K be a eld A sub eld of K is a subset L of K with the property that L is a eld under the same operations of addition and multiplication which are de ned for K Exercise 37 A subset L of a eld K is a sub eld ofK if and only if 1K 6 L ii ab E L implies a 7 b E L and iii ifu and U are nonzero elements in L then uuil E L De ne E to be an extension eld of F if there exists an injective ring homomorphism j F a E Lemma Let E be a eld and let j F gt E be an injectiue ring homomor phism Then a Im gt F is a sub eld ofE which is isomorphic to F b The map E FM a Final de ned by amoaiz Emmi a E F is a ring isomorphism Exercise 38 Proue this Lemma Proposition If f 220 aixi E is a polynomial of positiue degree ouer a eld F then there exists a eld E and an injectiue ring homomor phism j F gt E such that 220 ltjgta39vl 0 for some 1 E E Proof Assume the hypothesis By the theorem on page 13 f p1 p with r 2 1 and each p E irreducible Let p p1 and de ne E By Exercise 36 and the Proposition on page 13 of these notes E is a eld and the natural map 1 a E is a surjective ring homomorphism with kernel Let 1 F a E be the restriction of 1 to F ie gtu 1u u p for all u E F Then 1 is an injective ring homomorphism Since 1 pp2 197 E p Kerzz we have 1f f p 0E Hence 71 71 0 f p Za p 2 1990 29 i0 i0 De ne u x Then 1 E E and substituting we obtain 0 Mao gta1v gtanvm as claimed Using some elementary set theory and logic one can prove the following result see Hausen s Class Diary for MATH 6303 Spring 20057available upon request by email to hausenuhedu Lemma Let E and F be elds and suppose j F gt E is an injectiue ring homomorphism Then there exists a eld K with the following properties i F is a sub eld of K and ii there exists a ring isomorphism a K gt E such that 0a gta for all a E F This Lemma allows one to construct a eld K containing F as a sub eld in which the nonconstant polynomial 1 over F has a root 15 Theorem Given a polynomial f Ego aimi of positive degree over the eld F there exists a eld K containing F as a sub eld such that 0 for some w E K Proof Assume the hypothesis of the theorem Use the notation of the Proposition on page 14 and its proof7 and recall that the inverse of a ring isomorphism is a ring isomorphism Exercise 257 page 9 Then the Lemma implies that 0K ee HimW insemixa wr Since a 0a for all a e 10 0K ialaltaigtlta1ltvgtgti iailta1ltvgtgti fa 1v Hence w Ufa e K is a root of f The End

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