Math 3163 MATH 1100 - 009
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Date Created: 02/06/16
Theorems and Corollaries of Chapter 1 Well Ordering Axiom Every nonemptv subset of the set of nonnegative integers contains a smallest element Theoremlgi rhe Division Algorithm Let a and b be integers with b gt it Then there exists unique integers q and r such that o bq r and 0 g r C b Corollary 12 Let o and c be integers with c at ll Then there exist unique integers q and r such that a cq rand 0 S r lt lei Theorem 7113 Let a and b be integers not both 0 and let d geda 5 Then there exist not necessarily unique integers u and 1 such that d at l be Furthermore d is the smallest positive integer that can be written in the form out be Corollar 14 Let a and b be integers not both 0 and let i bea positive integer Then of gcda b if and only if 1 satisfies the following conditions l tile and dlb ii Ifclo and clb than old Theorem 15 Leta b and c be integers lfolbc and gedCa b 2 it then ale Theorem 16 TheEucl39ideon Algorithm Let a and b be positive integers with a 3 b if bl o then gcde b b If b l a then apply the division algorithm repeatedly as follows bimmn Osnltm r0 139quotqu r2 0 5 T2 lt1 r1 T1 T2 Q3 T3 T3 T2 This process ends when a remainder oiIO is obtained This must occur after a finite number of steps that is for some integer t T t a Tt lqt Tc 0 lt1 5 Ttsl reel TtQtsit 0 Then rt the last nonzero remainder is the greatest common divisor of q and b Theorems and 39Cerelilaries of Chapter 1 Lemma 17 Let a b qlr E Z and a t bq r then gedeb gedbr Theorem 18 Let b c and p be integers with p aquot D it Then p is prime if and onlv ifp has the property that whenever plbc then Me or plc Coroiigrg 19 lip is prime and plalag 21 then p divides at least one ofthe 1 Theorem 1153 Every integern except 0 i1 is the product of primes Theorem 111 The Fundamental Theorem ofArithmetid Every integer it except 1 i1 is the product of primes This prime factorization is unique in the fellowing sense if r1p1e2 quotpr and rlq1q2uqs with each pi and qj prime then r z s and after reordering and reiabei ing the qj s mzievmziw m teiw Coreiiar 11 Every integern gt 1 can be written uniquer in the form n pip mp1H where the pi are positive primes such that 331 3 32 S 1 pi Theorems and Corollaries of Chapter 3 Theorem 31 Let R and S be rings Define addition and multiplication on the Cartesian Product R x S by 733 r s r r s S and rsr s rr ss Then R x Sis a ring If both R and S are commutative then so is R X S If both R and 5 have an identity then so does R x 5 Theorem 32 Suppose that R is a ring and that Sis a subset of R such that i ii iii iv Sis closed under addition ie if a b E S then a b E S Sis closed under multiplication ie if a b E S then ab E 5 OR is in S and If a E S then the solution to the equation a x 2 OR is in S Then Sis a subring of R Theorem 33 For any element a in a ring R the equation a x 2 OR has a unique solution Theorem 34 If a b a c in ring R then b c Theorem 35 For any elements a and b in a ring R 1 2 3 4 5 6 aOROR ORa a b ab ab a a a b a b a b a b a b ab If R has an identity then 7 1Ra a Theorem 36 Let S be a nonempty subset of a ring R such that Sis closed under subtraction ie if a b E S then a b E S and Sis closed under multiplication Then Sis a subring of R Theorems and Corollaries of Chapter 3 Theorem 37 Let R be a ring and let a and b be in R Then the equation a x b has the unique solution x b a Theorem 38 Let R be a ring with identity and a and b elements of R If a is a unit then each of the equations ax b and ya b has a unique solution in R Theorem 39 Every field is an integral domain Theorem 310 Cancellation is valid in any integral domain R ie ifa at OR and ab 2 ac in R then b c Theorem 311 Every finite integral domain is a field Theorem 312 Let f R gt S be a homomorphism of rings Then 1 f0R 05 2 f a fa for every a in R 3 fa bfa fb forallaandbinR If R is a ring with identity and f is surjective then 4 Sis a ring with identity and f1R 15 5 Whenever u is a unit in R then u is a unit in S and fu391 fu391 Corollary 313 Iff R gt Sis a homomorphism of rings then the image off is a subring of S Theorems and Corollaries of Chapter 2 Theorem 21 Let n be a positive integer Then for any integers a b and c i a E amod n reflexive ii If a E b mod n then b E a mod n symmetric and iii If a E b mod n and b E c mod n then a E c mod n transitive Theorem 22 Let n be a positive integer Then for any integers a b c and d such that a E b mod n and c E d mod n we have i aCEbd modnand ii ac E bd mod n Theorem 23 Let n be a positive integer Then for any integers a and c we have that a E c mod n if and only if an cn Corollary 24 Two congruence classes modulo n are either disjoint or identical Corollary 25 Let n gt 1 be an integer and consider congruence modulo n i If a is any integer and r is the remainder when a is divided by n then a 1quot ii There are exactly 71 distinct congruency classes namely 0 1 n 1 Theorem 26 If a b and c d in Zn then a c b d and ac 2 bd Theorem 27 For any classes a b and c in Zn 1 If a and b in Zn then a Bb a b 2 al B bl Bch al BlbD Blcl 3 a b b a 4 aEB0 a 0ea 5 For each a in Zn the equation aEBX O has a solution in Zn 6 If a and b in Zn then a b ab 7 a b C a b C 8 a bEBC alCleDEB alCDlCD and aEBb C lal BlbDG alEBlCl 9 a b b a 10 a 1 a 1 a Theorems and Corollaries of Chapter 2 Theorem 28 If p gt 1 is an integer then the following statements are equivalent i p is prime ii For any a at O in Zp the equation ax 1 has a solution in Zp iii Whenever ab O in 22 then a O or b 0 Corollary 29 Let p be a positive prime For any a at O and any b in Zp the equation ax b has a unique solution in Zp Corollary 210 Let a b and n be integers with n gt 1 and gcdan 1 Then the equation ax b has a unique solution in Zn Theorem 211 Let a b and n be integers with n gt 1 and let gcdan d Then i the equation ax b has solutions in Zn if and only if d divides b and ii if d divides b then the equation ax b has exactly d distinct solutions in Zn Theorems and Corollaries of Chapter 4 Theorem 41 If R is a ring then there exists a ring P that contains an element x that is not in R and has these properties i Risa subring ofP ii xa ax for every 61 in R iii Every element of P can be written in the form 10 alx azx2 anxquot for some n 2 0 and a in R iv The representation of elements in P in iii is unique in the sense that if n S m 610 alx azx2 auxquot b0 blx bzx2 bmxm then a b fori S n and b OR for each i gt n v 10 alx azx2 anxquot OR if and only if a OR for all 139 Theorem 42 If R is an integral domain and fx and gx are nonzero polynomials in Rx then degfxgx degfx deg gx Corollary 43 If R is an integral domain then so is Rx Theorem 44 The Division Algorithm in Fx Let F be a field and f x and gx in F x with gx at 0F Then there exist unique polynomials qx and rx such that fx gxqx rx and either rx 0 or degrx lt deggx Theorem 45 Let F be a field and fx and gx in Fx not both zero Then there is a unique greatest common divisor dx offx and gx Furthermore there exist not necessarily unique polynomials ux and vx such that dx fxux gxvx Corollary 46 Let F be a field and fx and gx in Fx not both zero A monic polynomial dx in Fx is the greatest common divisor of fx and gx if and only if dx satisfies these conditions i dx divides both fx and gx ii if Cx divides both fx and gx then Cx also divides dx Theorem 47 Let F be a field and fxgx and hx in Fx lffx divides gxhx and fx and gx are relatively prime then fx divides hx Theorem 48 Let R be an integral domain Then fx is a unit in Rx if and only iffx is a constant polynomial that is a unit in R Theorems and Corollaries of Chapter 4 Corollary 49 Let F be a field Then fx is a unit in Fx if and only iffx is a nonzero constant polynomial Theorem 410 Let F be a field A nonzero polynomial fx is reducible in Fx if and only iffx can be written as the product of two polynomials of lower degree Theorem 411 Let F be a field and px a nonconstant polynomial in Fx Then the following conditions are equivalent 1 px is irreducible if bx and dad are any polynomials such that px divides bxcx then px divides either bx or 606 3 if rx and 306 are any polynomials such that px rxsx then rx or 06 is a nonzero constant polynomial Theorem 412 Let F be a field and px an irreducible polynomial in Fx If px divides a1xa2 x anx then px divides at least one of the a x Theorem 413 Let F be a field Every nonconstant polynomial fx in Fx is a product of irreducible polynomials in Fx This factorization is unique in the sense that if p1xl2x prx and Q1XQ2 q5x with each 1900 and qj x irreducible then r S After the appropriate reordering 1900 is an associate of q x for all 139 Theorem 414 The Remainder Theorem Let F be a field fx in Fx and a in F The remainder when fx is divided by the polynomial x a is fa Theorem 415 The Factor Theorem Let F be a field fx in Fx and a in F Then a is a root of the polynomial fx if and only if x a is a factor offx in Fx Corollary 416 Let F be a field and fx a nonzero polynomial of degree n in Fx Then fx has at most 11 roots in F Corollary 417 Let F be a field and fx in Fx with degfx 2 2 lffx is irreducible in Fx then fx has no roots in F Corollary 418 Let F be a field and fx in Fx with degree 2 or 3 Then fx is irreducible in Fx if and only iffx has no roots in F Corollary 419 Let F be an infinite field and fx and gx in Fx Then fx and gx induce the same function from F to F if and only iffx gx in Fx Theorems and Corollaries of Chapter 5 Theorem 51 Let F be a field and px a nonzero polynomial in Fx Then the relation of congruence modulo px is i reflexive fx E fx mod px for all fx E Fx ii symmetric iffx E gx mod px then gx E fx mod px iii transitive iffx E gx mod px and gx E hx mod px then fx E hx mod px Theorem 52 Let F be a field and px a nonzero polynomial in Fx Iffx E gx mod px and Mac E kx mod px then i fx hx E gx kx mod px ii fxhx E gxkx mod px Theorem 53 fx E gx mod px if and only if fx gx Corollary 54 Two congruence classes modulo px are either disjoint or identical Corollary 55 Let F be a field and px a polynomial of degree n in Fx Let S be the set consisting of the zero polynomial and all the polynomials of degree less than n in Fx Then every congruence class modulo px is the class of some polynomial in S and the congruence classes of different polynomials in S are distinct Theorem 56 Let F be a field and px a nonconstant polynomial in Fx f fx gx and M30 kx in F JG2906 then fx hx 906 C36 and fxhx gmC36 Theorem 57 Let F be a field and px a nonconstant polynomial in Fx Then the set Fxpx of congruence classes modulo px is a commutative ring with identity Furthermore Fxpx contains a subring that is isomorphic to F Theorem 58 Let F be a field and px a nonconstant polynomial in Fx Then Fxpx is a commutative ring with identity that contains F Theorem 59 Let F be a field and px a nonconstant polynomial in Fx Iffx E Fx and fx is relatively prime to px then fx is a unit in Fxpx Theorem 510 Let F be a field and px a nonconstant polynomial in Fx Then the following statements are equivalent i px is irreducible in Fx ii Fxpx is a field iii Fxpx is an integral domain Theorem 511 Let F be a field and px an irreducible polynomial in Fx Then Fxpx is an extension field of F that contains a root of px Corollary 512 Let F be a field and fx a nonconstant polynomial in Fx Then there is an extension field K of F that contains a root offx
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