∗ Algebra and Number Theory Lecture Notes Szymon Brzostowski November 1, 2020 General remarks on method and criteria of assessment (2020-2021): 1. The discussion class mark is the average of the marks from two or more tests. The mark may be increased in special cases (to students taking active part in the classes) up to one level up. 2. There will be one regular retake (scheduled just before or during the final exam- ination period) for those students who fail any of the aforementioned tests. 3. The lecture mark is the final exam mark (checking students’ theory knowledge), scheduled during the end-of-term examinations. 4. The final mark is comprised of the discussion class mark (50%) and the lecture mark (50%), provided that both are passing marks. 5. „Second sit” test/examination will take place during the resit examination period. Dates: end-of-term examinations (final examination period): 8–21 February 2021, resit (second sit) examination period: 1–7 March 2021 . This document has been written using the GNU TEXMACS text editor (see www.texmacs.org). 2 Algebra and Number Theory. Lecture Notes Szymon Brzostowski This text is intended to be a more detailed and formal version of the lecture itself. In order to get the most of it, I recommend reading it with your discussion-class notes at hand. Hopefully, this should allow for full comprehension of the material given. Szymon Brzostowski a Notation. The symbol „ ” will denote an assignment, for example means that the name a carries the value . More generally, „ ” can be treated as a defi- nition of a new concept (on its left side) using already known concepts (appearing on its right side). The phrase „if and only if” will be abbreviated as „iff”. Table of contents 1 Numbers and operations ................................. 3 2 Which numbers are „best”? ............................... 5 3 The first abstraction – the concept of a group .................. 7 4 ...Divisibility in Z ...................................... 9 5 ...Second abstraction – the concept of a ring .................. 11 6 The concept of an isomorphism ........................... 12 7 The direct sum ....................................... 15 8 The third abstraction – the concept of a field ................. 17 9 Congruences ......................................... 18 10 GCD ............................................. 20 11 Solving a · x ≡n b and a ·n x = b ........................... 23 12 The group Gn and the ϕ function ......................... 26 13 Fermat’s little theorem and its generalizations ................ 30 14 Chinese remainder theorem ............................. 35 15 The prime numbers ................................... 38 16 Systems of linear equations ............................. 42 16.1 Fromonetoseveralequations . ..... 42 16.2 Triangular systems and back substitution . ........ 45 16.3 Matrixnotationforalinearsystem . ...... 46 16.4 Rowechelonformofamatrix ........................ ... 50 17 The algebra of matrices ................................ 53 18 Which matrices are invertible? ........................... 57 19 Invertibility and determinants ........................... 62 20 Linear systems revisited ................................ 66 Further reading ......................................... 71 1. Numbers and operations 3 1 Numbers and operations We begin by setting the terminology for the basic number sets: f g the naturals (=positive integers): N , f g the integers: Z , o n p p q the rationals: Q Z N , q p p the reals: R; it is a set larger than Q; it contains e.g. the radicals ( p q 10 p p 10 e ) and also other numbers, like π π . fg Note that by convention N. The symbol N will denote the set N . There is one more set of numbers, easily constructed from R: fa b i a b g the complex numbers: C R . Here two complex numbers are added and multiplied similarly as polynomials, but further simplification of the i i i result is possible because the new number „ ” satisfies i . $ $ $ Clearly, we have the following chain of inclusions: N $ Z Q R C. Example. 1. Here’s how you add two complex numbers: i i i i i i . skip the brackets collect similar terms 2. And here is multiplication: i i i i i use the distributive law i i i i distributive law again i i . 2 use the relation for i collect similar terms Analyzing the first example above, it is not hard to produce a general rule for the addition of complex numbers: Addition Rule in C. a b i c d i a c b d i 8 . a b c d R (The universal quantifier is read „for all”.) A similar formula for the product of two complex numbers holds(try to find it yourself!): Multiplication Rule in C. a b i c d i a c b d a d b c i 8 . a b c d R 4 Algebra and Number Theory. Lecture Notes Szymon Brzostowski Observe that thanks to using symbols instead of specific numbers, above there are formulas that work for every choice of (real) values of a b c d. So although it is enough How for you to know how to calculate with complex numbers to arrive at the correct result, Algebra these formulas give you the result almost automatically once you remember them. In works algebra you generally work with symbols and you calculate using a certain set of rules for manipulating the symbols. Some of these rules you should already be familiar with. The basic rules for computations with numbers are , and – in symbols – b b a a (commutativity) (This law allows you to change the order of the summands.) , and – in symbols – b c a b c a (associativity) (This law allows you to skip all brackets when adding – the result is always the same regardless of which numbers you add first.) , and – in symbols – a b c a c b c (distributivity) (This law allows you to distribute multiplication over addition.) Note that the first two rules above are also true when multiplying numbers (i.e. with replaced by ). This suggests that perhaps such rules could be interpreted more abstractly. Namely, the symbol (or ) might denote an unspecified binary operation. Of course, in such generality it might be appropriate to use some new symbols (like , , , or ) for the binary operations and sometimes it is done so, especially when there is a danger of some confusion. Observe however, that in the case of complex numbers that we have considered, we used the ordinary symbols and , rather than some new ones, and we called them „addition” and „multiplication”, respectively. This is natural, because those operations are in fact extensions of the usual operations of addition and multiplication from the real numbers to a wider domain (this means that the „new” binary operations work in the „old” way for those complex numbers which i i i i i i are real, e.g. and similarly ). Also in computer science, a common practice is to implement some new binary operations and still denote them by and although in fact they overload or even totally redefine the standard operations of addition and multiplication. Example. One can define a truncated subtraction in N by the formula , if a b b a . b a b a , if There is everything in order with such definition, but you should feel that the usage of is a little cumbersome here (after all, we are rather subtracting than adding). In elementary arithmetic this truncated subtraction is usually denoted by „”. 2. Which numbers are „best”? 5 2 Which numbers are „best”? There is no unique answer to such question, because it depends on the properties you are interested in. For example, testing associativity and commutativity gives N Z Q R C Ass. YES YES YES YES YES + Com. YES YES YES YES YES , · Ass. YES YES YES YES YES Com. YES YES YES YES YES so in these respects there is no apparent difference between N, Z, Q, R and C. Actually, in principle, in order to verify that the entries in the above table are correct, it would be enough to check these properties for C because then they are automatically valid for the smaller sets N, Z, Q and R. In practice, this is done the other way around because of the order in which the number sets are actually constructed (N Z Q R C). As an example of such verification, try: Exercise 1. Check that + and are associative and commutative in C (assuming you know this for R!) using the formal definitions of + and given on page 3. Commentary. Proving the usual empirical properties of the operations and is not totally trivial, even for N. First, one needs to properly define the set N (this can be done using Set Theory) and the operations and . Only then is it possible to actually prove their properties (this is done in a branch of mathematics called Theoretical Arithmetic). Fortunately, you have so much empirical evidence for their validity from everyday life, you should not feel uncomfortably using those properties. Going back to our question again: are the various sets of numbers essentially different? Yes, they are... But only when you start thinking about solving equations x in them. Suppose we want to solve the equation in the naturals. Clearly, this is impossible because we have x N a result always greater than ! x But the same equation has a (unique!) solution x in Z. A difference! The following table shows which equations can be solved in the various number sets: Equation to solve N Z Q R C x b a NO YES YES YES YES x b a a NO NO YES YES YES . (1) x NO NO NO YES YES x NO NO NO NO YES x As we can see, Q is different than Z if considered with multiplication (e.g. x has no solutions in Z; on the other hand the equation does have the solution x Q C 10 in ). Also, is „better” than the other sets with respect to .