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9AN INTEGRATED APPROACH IN TEACHING ALGEBRA AND NUMBER THEORY: BEST PRACTICES
I.Subbotin, Professor,
National University, Los Angeles, USA,
L.Kurdachenko, Distinguished Professor and Chair, National University, Dnepropetrovsk, UKRAINE
У програмах навчання студент ¡в математичних спещальностей усюс ушверситет1в присутш обов 'язковг курсы лттног алгебры, абстрактног алгебри та теорп чисел. Усе бтьше стае очевидном, що студенты коми 'ютерних та Ыших спещальностей мають потребу у достатньо строгому викладг основ цих тръох дисциплЫ. У Америщ та захгдшй Сврот у бтьшт частит ушверсытепйв щ три дисциплЫи е суб'ектамы окремых лекцптых курсгв г для гх вывчення використовуються ргзт книги, прысвячеш кожшы з цих окремых дисциплЫ. Навчальш плани все бтьше розтягуютъся кожный раз, колы выныкае потреба выкладу нових застосуванъ. Базуючисъ на довготривалому досвгдг вгдомих укратсъких спецгалгстгв, автори аргументуютъ, що створення единого курсу, Ытегруючого щ три дисциплЫи, дастъ можливктъ бшьш ефективно та корисно використати вгдведений на це час. Цей об'еднуючий тдхгд буде допомагати бшьш глибокому розумЫню студентами гщх дисциплЫ г, разом з цим, кращому збергганню отриманих знанъ.
Algebra and Number Theory are two powerful, established branches of modern mathematics at the forefront of current mathematical research that are playing an increasingly significant role in many relatively new applications of mathematics such as computing, communications and cryptography. Historically, these two branches of mathematics have developed together enriching each other in the process. There is no way to draw a precise boundary separating these subjects. It would be appropriate to say that they actually form one common subject: Algebra and Number Theory. Thus, results in Number Theory are the basis and "a type of sandbox" for algebraic ideas and, in turn, algebraic tools contribute tremendously to Number Theory. It is interesting to note that such newly developed branches of mathematics as Coding Theory heavily use ideas and results from both Linear Algebra and Number Theory.
Linear Algebra, Abstract Algebra and Number Theory are three mandatory disciplines in all university mathematics
majors programs around the globe that every student should take. Increasingly, it is also becoming evident that students of Computer Science and other such disciplines also need a strong background in these three areas. In American and West European universities most of the time, these three disciplines are the subject of different and separate lecture courses that use different books dedicated to each subject individually (see, for example, such classical books as [1-8]). In a curriculum that is increasingly stretched by the need to offer traditional favorites, while introducing new applications, we think that it is desirable to introduce a fresh approach to the way these three specific courses are taught. Based on time honored experience of teaching these subjects in Ukrainian and Russian universities, we argue that one course, integrating these three disciplines, together with a corresponding book for this integrated course, would be helpful in using class time more efficiently. As an argument supporting this statement, we mention that many theorems in Number Theory have very simple proofs using
algebraic tools. Most importantly, we think the integrated approach helps to build a deeper understanding of the subject in the students, as well as improve their retention of knowledge. In this respect, the time-honored European experience of integrated Algebra and Number Theory courses, organically implemented in the university curriculum, is very efficient.
As an example, we expose below a detailed outline for the lecture dedicated to Arithmetic Functions. One of the most important reasons for this is to illustrate the capabilities of giving a systematic, integrated and complete description of the theory of the main number systems that form a basis for the structures that play a central role in various branches of mathematics. We think that such an approach is appropriate for students in computer science or mathematics who possess a certain degree of general mathematical knowledge pertaining to typical students at this stage. Of course we assume that the student is comfortable with mathematical formalism and also has some experience in reading and writing mathematical proofs. We emphases perspective here will no doubt be different from those in a traditional mathematical presentation of these subjects at American universities. However the proposed way of exposition of material has been successfully class tested during decades of teaching at many Ukrainian and Russian Universities (this is enough to mention such classical books as [9-11] and a modern book [12]). As a typical example of the integrated approach we attach the detailed outline of the lecture dedicated to Arithmetic Functions.
Arithmetic Functions lecture outline
We consider some important number-theoretic functions that are the functions whose domain is the set of natural numbers, i.e. any function
f: n->c.
A number-theoretic function whose range is included in the set of complex numbers is called an arithmetical function or an arithmetic function.
We list now some important examples of number-theoretic functions.
Let v: N — N be the function defined in the following way. Put v(1) = 1 and if n > 1, then v(n) is the number of all positive divisors of n.
We define also the function c N — N by the rule. Put c(1)=1 and if n>1, then c(n) is the sum of all positive divisors of n.
The next proposition, the proof of which is standard and well known, provides us with the formulas for the values of these functions.
Proposition 1. Let n be the positive integer and suppose that
n = pik1 p2k2 . . .ptkt where pj is a prime, 1^j<n, and pk # pj whenever k ^ j. Then
v(n) = (k1+1). . .(kt + 1), and c(n)=((p1 k1+ 1-1)/(p1 - 1))...((ptkt+1 - 1)/(pt -1)).
There is a very interesting number-theoretical problem connected to the function y(n). A positive integer n is called perfect, if y(n) = 2n. For example, the positive integers 6 and 28 are perfect. Proposition 1 implies that if 2k + 1 - 1 is a prime, then n = 2k (2k + 1 - 1) is perfect. L. Euler has proved that every even perfect number has a such form. Thus the problem of finding all perfect numbers is reduced to the finding primes of the form 2k + 1 - 1.
Recall, that a prime p is called Mersenne prime if p=2k + 1 - 1 for some positive integer k.
The following two important problems about perfect numbers are unsolved at the time of writing:
1. Are there infinitely many perfect numbers?
and
2. Is there an odd perfect number?
We want to construct further number-theoretic functions from given ones. They are many ways to do this but one particularly interesting method is as follows:
The Dirichlet product of two number -theoretic functions f and g is the function fDg, defined by the following rule (fDg)(n) = Z kt = n f(k)g(t).
Next we expose to students the well known proof of the following statement.
Proposition 2. (i) Dirichlet multiplication of number-theoretic junctions is commutative.
(ii) The Dirichlet multiplication of number-theoretic functions is associative.
(iii) The Dirichlet multiplication of number-theoretic functions has an identity element. This
is the function < defined by the rule <(1)= 1 and
<(n) = 0 for n > 1.
Another important number-theoretic function is the function S defined by the rule S(n)=1 for each neN. We have
(f D S)(n) = I kt=n fk) S(t) = I k I n fk) for each neN (we take the sum on all divisors k of the number n).
The function fD S is said to be the summator function for a function f.
The next important number-theoretic function that we consider is the M^ius function [ defined by the rule: №) = 1;
if n > 1 and n = p1 1 p2 2...pt t where p1, p2, ... pt are primes and pi #pj whenever i #j, then if there exists j such that kj > 2, then
[n) = 0; and,
if n = p1 p2...pt, then [n) = (-1). The following proposition could be proven by ordinary induction.
Proposition 3. [D S = <.
Based on this result the instructor is ready to prove the following famous
Theorem 1 (the Mubius inversion
formula). Let f be a number-theoretic function and F be the summator function for a function f. Then
f(n) =Z kin M(k)F(n/k) for each neN.
The next number - theoretic function plays a very important role in many domains of mathematic. It is the Euler function 9, which defined by the following rule:
91) = 1,
if n > 1, then (p(n) = I I where 0n = {k Ik eN, 1 <k < n, GCD(n, k) = 1}.
The next theorem provides us with an alternative approach to the Euler function. This shows one important use of this
function in group theory. We will expose it with the proof.
Theorem 2. (i) Let G be the finite cyclic group of order n. Then the number of all generators for G coincides with pp(n).
(ii) If k is a positive integer, then k + nZ e U(Z/nZ) if and only if GCD(k, n) = 1. In particular, ( (n) = iu(Z/nZ) i.
Proof. (i) Since G is cyclic, G = < g > for some element geG. It is well known that an element y = gk e G is a generator for G if and only if GCD(k, n) = 1. So we have < g > = { g0, g1, g2, ... , gn - 1}. This shows that we can choose a number k such that 1 < k < n. It follows that p(n) is equal to the number of the generators for a group G.
(ii) Let k + nZ e U(Z/nZ). Then there exists a coset s + nZ such that
(ks + nZ) = (k + nZ)(s + nZ) = 1 + nZ. In other words, ks + nr = 1 for some re Z and (k, n)=1. Conversely, if GCD(k, n)=1, then repeating the previous arguments in converse order we obtain that k + nZ e U(Z/nZ). It implies the equality I U(Z/nZ) I = p(n).
Corollary (Euler's Theorem). Let n be a positive integer. Supose that k be an integer such that GCD(k, n)=1. Then kppn)=1(mod n).
Corollary (Fermat's little theorem). Let p be a prime and k be an integer. If p does not divide k, then k - 1 = 1 (mod p).
The next step of the lecture is to find the summator function for the Euler function. We will do it with the help of some elementary group theoretical results.
Let G be a group. Define the binary relation o on G by the following rule: x o y if and only if | x | = | y |, x, y e G.
It is not hard to check that o is an equivalence relation. If n is a positive integer, then put
Gn = {x | x e G and | x | = n}. The subset Gn is the class of equivalence by the relation o. Denote by G« the subset of all elements of G whose orders are infinite. From elementary group theory results it follows that the family of subsets
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{Gn \ n e Nu {«}} is a partition of the G. Observe that the subset Gn can be empty for some positive integer n. In particular, if G is a finite group then G« = 0. Moreover, ify is an arbitrary element of a finite group G, then | y | is a divisor of \ G \. Thus a family of subsets {Gk \ k is a divisor of \ G \} is a partition of the finite group G. It follows the equation G = u k\n Gk where \ G \ = n.
Note that in majority of cases some subset Gk can be empty. As we can see, the only one exception is a cyclic group (in other words, Gk is not empty for all k | |G| if and only if G is a finite cyclic group).
These observations allows us to obtain the following interesting identity.
Theorem 3. Zk /n (k) = n.
Proof. Let G=<g> be a cyclic group of order n. We have observed above that G = u k\n Gk. Choose an arbitrary divisor k of the number n and put d = n/k and x = gd. We have xk = (gd)k= gdk = gn = e. If we suppose that x* = e for some positive integer t < k, then e = xt =(gd)t = gdt =e. Since dt < n, we obtain a contradiction with the fact that | g | = \ G \ = n. Thus | x | = k. In other words, it proves that the subset Gk is not empty for every divisor k of n.
Let z be the element of G having of order k. Then z = gm for some positive integer m and e = zk = (gm)k = gmk. It follows that n = dk\mk and hence d divides m, that is, m = ds for some positive integer s. We have
„m ds /„d\s s z = g = g = (g) = x
It proves that z e < x >.
Furthermore | < z > | = | z | = | x | = k, so that < z > = < x >. In other words, every element of order k is a generator for a subgroup < x >. The number of all such elements is equal to ((k). Hence for each divisor k of a number n and we have \ Gk \ = ((k). Therefore
n = \G\ = I k\n \ Gk \ = I k\n ((k).
We will omit the proofs of the following corollaries here.
Corollary. Let G be a finite group of order n. If k is a divisor of n then put G[k] = {x ¡x e G and ¡x = e }. Suppose that /G[k] / <k for each divisor k of n. Then the group G is cyclic.
Corollary. Let F be a field and G be a finite subgroup of U(F). Then G is cyclic.
Corollary. Let F be a finite field. Then its multiplicative group U(F) is cyclic.
Corollary. Let p be a prime. Then U(Z/pZ) is cyclic.
We have showed above that the identity permutation of a set N is the summator function for Euler function. This allows us to obtain a formula for values of the Euler function. Employing Theorems 1 and 3 we have
((n) = I k\n |(k) (n/k) = n I k\n |(k)/k for each n e N. Let now n = p1k1 p2k2...ptkt be the primary decomposition of n, where p^p whenever i ^ j. If m is a divisor of n, then
m = pisi p2s2... ptst, where 0 < sj < kj, 1<j<t. If there exists j such that sj > 2, then |(m) = 0. Hence
I k\n |(k)/k = 1 -1 1 <j <t (1/pj) + Il <j <m<t (1/pj pm) -... + (-1)t (1/p1p2...pt) = (1 - (1/p1)) (1 - (1/p2)) .. (1 - (1/pt)). Consequently, ((n) = n(1 - (1/p1)) (1 - (1/p2))...(1 - (1/pt)) = =p1k1 - 1(p1 - 1))p2k2 -1(p2 - 1)...ptkt - 1(pt - 1). In particular, if p is a prime, k e N, then
((pk) = pk - 1(p - 1) = pk - pk -1. Moreover, from the obtained above formula we can derive that ((nk) = ( (n) ((k) whenever GCD(k, n) = 1. The number - theoretic function fis called multiplicative, if it satisfies the following conditions:
there is a positive integer n such that f(n)?0; if GCD(k, n ) = 1 then f(nk) = f(n) f(k). The next theorem supplies us with an important property of multiplicative functions.
Theorem 4. If the number-theoretic junction fis multiplicative, then the summator function for fis also multiplicative.
Proof. Let k, t be the positive integers such that GCD( k, t ) = 1. If d is a divisor of kt, then clearly d = uv, where u \ k, v \ t. Let F = f D S) be the summator function for f Then
F(kt) = I u\k, v\t f(uv) = I u\k, v\t f(u)f(v) =
I u\kfu) =I v\t f(v) = F(k)F(t). At the end of this lecture, we ususally discuss some applications of the above results. Recently the Euler function finds its applications in cryptography. A giant leap forward occurred in cryptography in the
second half of the twentieth century, with the invention of public key cryptography. The main idea is the concept of a trapdoor function-- a function that has an inverse, but whose inverse is very difficult to calculate. In 1976, R L Rivest, A.Shamir, and L.M.Adleman succeeded in finding such a class of functions. It turns out that if you take two very large numbers and multiply them together, a machine can quickly compute the answer. But, if you give the machine the answer and ask it for two factors, the factorization will not appear in a useful amount of time. The public key system, built upon these ideas, is now known as RSA-key after the three men who created it. In classic code tuples the method of coding is the main secret. But in the tuples with the open key there is no sense to keep such a secret. The main idea of RSA-coding is in the following. One chooses two arbitrary primes p and q and calculates n=pq and p(n)=(p-1)(q-1). Next, one picks an arbitrary number k<p (n) that is relatively prime with p (n). As we can see by the proof of Theorem 2, k + p(n)Z e U(Z/p(n)Z). Consequently, there exists a positive integer t such that kt = 1(mod p (n)). We can find a number t with the help of the Euclidian algorithm. The numbers n and k determine the coding method. They are not secret and they form the open (or public) key. Only the primes p, q and the number t are kept in secret. First, the massage should be written in numerical form with the help of ordinary digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. After that, one divides this message into blocks M1, ..., Ms of length w. The number m should satisfy the restriction 10w < n. Usually one chooses the numbers p and q up to 100 digits each. Each block Mj can be considered as a representative of the coset Mj +n Z. The codling of the block Mj is in the substitution of it by the block E(Mj) such that E(Mj)= Mj k(mod n). Since the number n is big enough, only a computer can perform it. The decoding is done by the following procedure. The choice of the number t requires that kt = 1+rp (n). Now one can apply Euler's theorem. However, in this case Mj and n are required to be relatively
prime. Nevertheless, we can show that this application is valid in any case. Let m be an arbitrary positive integer. If GCD(m,n)=1, then mp (n) = 1(mod n), and
mkt = m1+rp(n) = m(mp(n))r = m( mod n ).
Suppose now that GCD(m,n) f 1. Since n=pq, then either p divides m and q does not divide m, or conversely, q divides m and p does not divide m. Consider the first case; the consideration of a second case is similar. We have m = pu and mkt - m = (pu)kt - pu.
On the other hand, mkt = m1+I(n) = m1 +«p-1 ^-1) = m (m(q-1)^-1).
Since GCD(m,q)=1, the second corollary from Theorem 2 led us to (m(q - 1)) = 1 (mod q). Therefore, mkt = m (m(q - 1)) ^ - 1) (mod q) = = m (1(q - 1)) r(p - 1) (mod q) = m (mod q).
Hence q divides mkt - m. As we have seen above, p divides mkt - m, so that n=pq divides mkt - m. Consequently, in any case we have mkt = m(mod n). For block Mj we have E(Mj)t = (Mj k)t = Mjkt = Mj1 + Ip(n) = Mj (Mj(n))r .
So we can write E(Mj)t = Mj (mod n). Recall that the number Mj satisfies the condition 1<Mj<n, and therefore it is uniquely determined by the congruence EMj)=Mj (mod n).
The problem of reliability of the RSA-code is reduced to the question: Can the block E(Mj) be decoded? For this one needs to solve the congruence xk = E(Mj)(mod n) without knowing the number t. Soon we will study congruences of the type xk = a (mod n). By now, we can only state that there is no general method for the solution of such congruences. In reality, it could be done by the examination of all cases. From the choice of w it follows that this sorting requires the consideration of 100100 cases, which is not quite realistic.
Therefore, to crack the RSA-code one needs to find t if n and k given. If the decomposition n = pq is known, then this is not difficult. In turn, knowing the numbers n, k, and t, one can find the decomposition n = pq. So the finding of t requires the same efforts as is needed for finding the decomposition n = pq. However, in the case when each of the factors has 100 digits this is not realistic yet.
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1. Dummit, D., Foote, R., Abstract Algebra, Hoboken: Wiley, 2007.
2. Beachy J.A. and Blair W.D., Abstract Algebra, Illinois: Waveland Press, 1996.
3. More, T. Elements of Abstract Algebra, the University ofMichigan: Macmillan, 1967.
4. Lay, D.C., Linear Algebra and Its Applications, New York: Addison Wesley, 2002.
5. Campbell, H.G., Linear Algebra with Applications, Boston: Prentice-Hall, 1980.
6. Niven, I.M., Zuckerman, H.S. An Introduction to the Theory ofNumbers, Hoboken: Wiley, 1972.
7. Ireland K.F., Rosen, M.I. Classical Introduction to Modern Number Theory, Berlin: Springer, 1982.
8. Hasse, H., Zimmer, H.G., Number Theory, Berlin: Springer, 2002.
9. Куликов Л.Я. Ачгебра и теория чисел. Москва: Высшая школа, 1979.
10. Курош AT. Курс высшей ачгебры. Наука: Москва, 1971.
11. . ., . ., теория чисел. Москва: Просвещение, 1974.
12. Курдаченко Л А., Кириченко В. В., ..
meopii чисел, Ku'ie: 1нститут Математики НАНУ, 2005.
Резюме. Subbotin I., Kurdachenko L. AN INTEGRATED APPROACH IN TEACHING ALGEBRA AND NUMBER THEORY: BEST PRACTICES. There are three mandatory courses, Linear Algebra, Abstract Algebra and Number Theory, in all university mathematics programs that every student of mathematics should take. Increasingly, it is also becoming evident that students of Computer Science and other such disciplines also need a strong background in these three areas. In America and West Europe most of the time, these three disciplines are the subject of different and separate lecture courses that use different books dedicated to each subject individually. In a curriculum that is increasingly stretched by the need to offer traditional favorites, while introducing new applications. Based on the time honored Ukrainian experience, the authors argue that one course integrating these three disciplines would be helpful in using class time more efficiently. This integrated approach will help build a deeper understanding of the subject in the students, as well as improve their retention of knowledge.
Summary. Subbotin I., Kurdachenko L. ИНТЕГРИРОВАННЫЙ ПОХОД К ОБУЧЕНИЮ АЛГЕБРЕ И ТЕОРИИ ЧИСЕЛ: ЛУЧШИЕ ДОСТИЖЕНИЯ. В программах обучения студентов математических специальностей всех университетов имеются обязательные курсы линейной ачгебры, абстрактной ачгебры и теории чисел. Все больше становится очевидным, что студенты компьютерных и других спегщачьностей нуждаются в достаточно строгом изложении основ этих трех дисгщплин. В Америке и западной Европе в большей части университетов эти три дисциплины являются субъектами отдельных лекционных курсов и для их изучения используются различные книги, посвященные каждой из этих отдельных дисгщплин. Учебные планы все больше растягиваются каждый раз, когда возникает потребность изложения новых применений. Базируясь на длительном опыте известных украинских спегщачистов, авторы утверждают, что создание единого курса, интегрирующего эти три дисгщплины, позволит более эффективно и полезно использовать отведенное на это время. Этот объединяющий подход будет способствовать более глубокому пониманию студентами этих дисгщплин, а заодно и лучшему сохранению полученных знаний.
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