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# Class Note for MATH 294A with Professor Savitt at UA

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 20 views.

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Date Created: 02/06/15
Algebra Math 294A Problem Solving Seminar 1 Polynomials One of the most important things to know when solving a problem involving polynomials is how to factor There are a large number of identities which can be useful when factoring7 but the most basic ones are an 7 bn a7 ban71 an72b I I 39 abn72 bn717 a b a ba 1 7 a 2b 7 ab 2 Innil7 n odd Similar to division and factoring in the integers7 we may apply the Division algorithm to polynomials Division Algorithm Let and 91 be either polynomials with coefficients in R C or Q or monic poly nomials over Z Then there exist unique polynomials of the same type 41 and 117 such that fI 4Iyr MI where deg39r1 lt degg17 and 91 divides precisely when 11 is the zero polynomial Like in the integers7 the division algorithm can be used to nd the greatest common divisor of two polynomials Example 1 Let 11 and 12 be the roots of the equation 12 7 ad1 ad7bc 0 Show that 1 and 1 are the roots of the equation y2 7 a3 d3 Saba 3bcdy ad 7 be3 0 Example 2 Prove that the fraction n3 2nn4 3n2 l is irreducible for every natural number n Example 3 Let N be the number which consists of 91 consecutive 17s in base ten expansion Prove that N is composite Problem 1 Show that n4 7 20n2 4 is composite for any integer n Problem 2 Determine all solutions in the real numbers 17 y z w of the system zy2w7 1I1y1z1w Problem 3 For What it is the polynomial 112z4 z2 2 divisible by the polynomial lz12 z 1 Problem 4 Consider all lines Which meet the graph of y2147133175 in four distinct points7 say ziyi i 1234 Show that 11I213I4 4 is independent of the line7 and nd its value Problem 5 Prove that there are no prime numbers in the in nite sequence of integers 100011000100011000100010001 H Problem 6 Given numbers zy2 such that zy23 12y222 5 13y323 77 nd the value of I4 y4 24 Problem 7 If n gt 17 show that z l 7 I 7 l 0 has a multiple root if and only if n 7 l is divisible by 6 Problem 8 Let 131 be the following polynomial7 With real coef cients 131 anzn an71zn 1 agz3 12 1 1 Where n 2 2 Show that the equation 131 0 cannot have all real roots 2 Groups Let S be a set A binary operation on S is a function from S X S to S For example7 addition is a binary operation on Z A binary operation 96 on S is associative if r 96 s 96 t r 96 s 96 t for all rst E S A group is a nonempty set C With an associative binary operation 96 such that i G contains an identity element7 e E G Which has the property 6 96 g g 96 e g for every 9 E G ii G contains inverses of elements That is7 for every 9 E G there is an element h E G such that g 95 h h 96 g 67 Where e is the identity element of G So7 for example7 Z With addition is a group Note that the operation is not required to be commutative An example of a group With a noncommutative operation is the group of tWObytwo matrices over R With multiplication In an arbitrary group the binary operation is often denoted by juxtaposing two elements as in multiplication so that ab means the operation performed on a and b The theory of groups is a huge subject so we restrict ourselves to only a few facts and notions Firstly the identity element and inverses in groups are unique prove this as an exercise Secondly groups have the left and right cancellation property That is for all a bc E G ab ac implies b c and ab cb implies a c A subgroup of a group G is a subset of G which forms a group under the same operations as H For example the set of even integers is a subgroup of Z under addition Example 4 Let a and b be two elements in a group such that aba ba2b a3 e and b2n 1 e for some positive integer n Prove that b e where e is the identity element Problem 9 Let G be a set with an associative binary operation such that for all ab E G a2 b ba2 Show that G is a group and the operation is commutative Problem 10 Let A be a subset of a nite group G such that A contains more that half of the elements of G Prove that each element of G is the product of two elements of A Problem 11 Prove that no group is a union of two proper subgroups that is subgroups which are not the group itself Problem 12 Let S be a nonempty set with an associative binary operation with the left and right cancella tion properties Assume that for every a E S the set a l n 123 is nite Must S be a group

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