Fundamental concept of algebra pdf

The book then proves the Schmidt-Ore theorem, and also describes linear algebra, as well as the Birkhoff-Witt theorem on Lie algebras. The text also addresses ordered groups, the Archimedean groups and rings, and Albert's theorem on normed algebras.

This book can prove useful for algebra students and for professors of algebra and advanced mathematicians. For nearly 50 years, A. Albert thrived at the University of Chicago, one of the world's top centers for algebra.

His "pure research" in algebra found its way into modern computers, rocket guidance systems, cryptology, and quantum mechanics, the basic theory behind atomic energy calculations. This first-hand account of the life of a world-renowned American mathematician is written by Albert's daughter.

Her memoir, which favors a general audience, offers a personal and revealing look at the multidimensional life of an academic who had a lasting impact on his profession.

There are, instead, many bad teachers and bad curricula ""The difficulty of learning mathematics is increased by the fact that in so many high schools this very difficult subject is considered to be teachable by those whose major subject is language, botany, or even physical education.

This doesn't mean that the researcher is controlled by them. Many go their own way, ignoring the fashionable. That's part of the strength of a great university. This book aims to make the general theory of field extensions accessible to any reader with a modest background in groups, rings and vector spaces.

Galois theory is regarded amongst the central and most beautiful parts of algebra and its creation marked the culmination of generations of investigation. A substantial amount of this book is devoted to general questions including significant material from the history of science, allowing one to follow the formation of modern attitudes on the essence of mathematics and the methods of its applications : only chapters 5 and 6 are devoted to a survey of the basic algebraic structures and a more detailed analysis of a structure associated with some geometric considerations, are of a more concrete character.

The Fundamentals of Mathematical Analysis, Volume 1 is a textbook that provides a systematic and rigorous treatment of the fundamentals of mathematical analysis. Emphasis is placed on the concept of limit which plays a principal role in mathematical analysis. Examples of the application of mathematical analysis to geometry, mechanics, physics, and engineering are given. This volume is comprised of 14 chapters and begins with a discussion on real numbers, their properties and applications, and arithmetical operations over real numbers.

The reader is then introduced to the concept of function, important classes of functions, and functions of one variable; the theory of limits and the limit of a function, monotonic functions, and the principle of convergence; and continuous functions of one variable. A systematic account of the differential and integral calculus is then presented, paying particular attention to differentiation of functions of one variable; investigation of the behavior of functions by means of derivatives; functions of several variables; and differentiation of functions of several variables.

The remaining chapters focus on the concept of a primitive function and of an indefinite integral ; definite integral; geometric applications of integral and differential calculus.

This book is intended for first- and second-year mathematics students. This is the revised edition of Berlekamp's famous book, 'Algebraic Coding Theory', originally published in , wherein he introduced several algorithms which have subsequently dominated engineering practice in this field. One of these is an algorithm for decoding Reed-Solomon and Bose-Chaudhuri-Hocquenghem codes that subsequently became known as the Berlekamp-Massey Algorithm.

Another is the Berlekamp algorithm for factoring polynomials over finite fields, whose later extensions and embellishments became widely used in symbolic manipulation systems. Other novel algorithms improved the basic methods for doing various arithmetic operations in finite fields of characteristic two. Other major research contributions in this book included a new class of Lee metric codes, and precise asymptotic results on the number of information symbols in long binary BCH codes.

Selected chapters of the book became a standard graduate textbook. Both practicing engineers and scholars will find this book to be of great value. This book contains the collected works of A. Adrian Albert, a leading algebraist of the twentieth century. Albert made many important contributions to the theory of the Brauer group and central simple algeras, Riemann matrices, nonassociative algebras and other topics. How to make your Fundamentals of Algebra strong? Students who have Algebra as part of their curriculum Any individual who loves the subject and wants to be thorough with the basics of Algebra Parents or teachers who are looking for an additional help with Algebra.

Quiz video helps in testing your knowledge. Fundamentals Of Algebra Textbook In short it is an interesting course fulfilling all the student's needs. Anyone who has Algebra as a subject in their curriculum.

Work through the following problems prior to reading further. Derive the consequences of Lemma 2. Are there any integers m that cannot be written in this form? Suppose you want to put a certain number of gallons of water into a pool, and you only have an 8 gallon bucket and a 5 gallon bucket with no markings to measure the water. How could you measure 7 gallons of water into the pool?

What about 11 gallons? For any positive integer m, how could you measure m gallons into the pool? Suppose now that you have only a 9 gallon bucket and a 12 gallon bucket. Could you measure 15 gallons of water into the pool?

What about 20 gallons? What integer numbers of gallons of water can be measured? What relationship between the bucket sizes is necessary in order to guarantee that every positive integer number of gallons of water can be measured with two buckets? Given sufficient water supply, of course. Why or why not? Do these calculations suggest a general result? Do these examples suggest a general result?

We now study further properties of the GCD and derive some consequences of previous theorems in the case that a and b are relatively prime. This is Theorem 2. We were able to describe all integers that can be expressed in this form. We get the converse of Theorem 2. Therefore, by Theorem 2.

We do get these implications if we add an appropriate condition, however. The proofs are nearly identical to those of Theorems 2. For those that can, find r and s, and for those that cannot, explain why. Explain how you could measure out exactly 13 ounces of water given a 7 ounce cup and a 9 ounce cup. Is there any positive integer m for which it would be impossible to measure m ounces of water? Assume an unlimited supply of water. Which of the amounts of water below could be measured given a 6 ounce cup and a 9 ounce cup, and which could not?

Fully explain your answer. What is a prime number? What is a composite number? Is 1 a prime? Are there any even primes? If so, how many? How can we determine if a number is prime or not? That is, given an integer n, how would you go about deciding whether or not n is prime? Is it necessary to check all positive integers less than or equal to n as potential divisors?

How many primes are there? Finitely many? Infinitely many? Is there a largest prime number? We have seen that in general, if a bc then a may or may not divide b or c. If p is prime and p bc, does p necessarily divide a or b? Can you find an example where p is prime and p bc, but p - b and p - c? If p is prime, can p be a rational number? If n is a positive integer, under what conditions is n a rational number? Can n be rational and not be an integer?

What is meant by the prime factorization of a positive integer? Can every positive integer be written as a product of prime numbers? Can a positive integer be written as a product of primes in more than one way other than the obvious variation of changing the order of the primes?

The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, The only even prime is 2, since every even number has 2 as a factor. The websites www. By the Well-ordering Principle Theorem 2. Thus p is the smallest divisor of n that is greater than 1. We will show that p is a prime. Suppose p is not prime. Thus by transitivity of divisibility Theorem 2.

Therefore, our assumption that p is not prime must be false, and so p is prime. By the Prime Divisor Principle, it is only necessary to determine whether each prime less than n is a divisor of n. The next results further reduce the amount of work required.

Hence a 6 n. By the Prime Divisor Principle, there is a prime p such that p a. Since p a and a n, Theorem 2. If no prime p with p 6 n divides n, then n is prime. If n were composite, then Lemma 2. By hypothesis, this is not the case, and so n is prime. First write down all the numbers from 2 up to n. Start by crossing out all multiples of 2. The first number not crossed out is a prime in this step, 3.

Next cross out all multiples of this next prime. Again, the next number not crossed out is prime. All of the numbers not crossed out are then primes. Unfortunately, this method has several drawbacks.

What possible drawbacks to this method do you see? However, there can be no absolute largest prime, as the following result of Euclid implies. Suppose that there are only finitely many primes, say p1 , p2 , p3 ,. Hence our assumption that there are only finitely many primes must be false. Hence either p a or p b. We proceed by induction on the number n of factors. Let p be prime and suppose p is rational. As before, this implies p b2 , and so by Theorem 2.

We have shown that if n is rational, then n is an integer, and the theorem follows. The expression of n as a product of primes is unique except for the order in which the factors are written.

We first show such a factorization exists. By the minimality of m, each of a, b is either prime or a product of primes. Therefore, our assumption that there is an integer greater than 1 that is neither prime nor a product of primes must be false. By Corollary 2. The theorem says that although we can rearrange the order of the product any way we like, every expression of , as a product of primes will involve the primes 2, 3, 5, 7, and no others, and there will be four factors of 2, two factors of 3, one factor of 5, and three factors of 7.

Use the Prime Test Theorem 2. Show that a 2-digit number n is prime if and only if n is not divisible by 2, 3, 5, or 7. Use the Sieve of Eratosthenes to find all primes less than Find the canonical prime factorizations of the following integers. Prove the following. Without doing any calculations, explain why a right triangle cannot have sides with lengths 2, 3, and 3. Could there be a right triangle with sides of lengths 2, 3, and 3. Show that if a right triangle has two sides of integer length, then the length of the third side is either an integer or is irrational.

Find the prime factorizations of some non-divisors of For example, find the prime factor- izations of 16, 27, 32, and 54, which are all divisible by the same primes as 72, and the prime factorizations of 15 and Compare the exponents in the prime factorizations of divisors and non-divisors of 72 with those in the factorization of Use the examples above to make a conjecture about the conditions on the exponents that are required in order that a b.

How many positive divisors does 72 have? Write down all divisors of with their prime factorizations. How many are there? Can you use these examples to guess a formula for the number of divisors of an integer in terms of its prime factorization? What is meant by the least common multiple of two integers a and b?

Do you know or can you derive a formula for the greatest common divisor and least common multiple of a and b in terms of the prime factorizations of a and b? Why does the formula work? In the canonical prime factorization of an integer, we require that all of the exponents be positive, otherwise the factorization would not be unique. When comparing two integers, however, it is often useful to allow the exponents to be zero so both integers can be written as products of powers of the same primes.

Thus the distinct primes dividing a or b are 2, 3, 7, and Notice that a does not have a factor of 11 and b does not have a factor of 3. Then a b if and only if ai 6 bi for all i. If p is any prime divisor of n, then p n and n b, hence p b by Theorem 2. Counting the number of possible combinations of exponents yields the following result. The number of positive divisors is the the total number of possible combinations of exponents, which is the product of the number of choices for each exponent, i.

A concept closely related to the greatest common divisor is the least common multiple. Using the characterization of divisibility above, we can express both the GCD and LCM of a and b in terms of prime factorizations. Then a. Therefore, it follows from Theorem 2. If p is any prime divisor of c, then by Theorem 2.

Since c a and c b, Theorem 2. Thus c 6 d and ii holds. Each prime divisor of a or b will also divide c, but c may have other prime divisors as well. Similarly, any common multiple of two integers is a multiple of the LCM. If c is any integer satisfying a c and b c, then m c. This is shown in part ii for the LCM in the proof of Theorem 2. We use the notation of Theorem 2. Hence di is one of ai or bi and mi is the other. The Euclidean algorithm can be used to find a, b , and then this formula allows us to calculate [a, b] easily.

Determine whether each of the following integers divides n. Explain your answers. Determine the number of positive divisors of the following integers. Do not factor the integers.

Explain your answer. Do not factor. Thus, for a fixed integer n, the remainder on division of m by n is uniquely determined by m. We can therefore classify all integers according to their remainder on division by n, putting two integers in the same category if they leave the same remainder on division by n.

Every integer leaves a remainder of 0, 1, or 2 on division by 3. The following result gives an easier characterization of this idea. The integer n in the definition is called the modulus, and will always be a positive integer. The concept of two numbers being congruent is only of interest if the numbers are integers. Note that the least non-negative residue of a modulo n is simply the remainder on division of a by n. For example, 3 is the least non-negative residue of 55 and of 29 modulo 13, as seen in the example above.

The next theorem says that congruence is an equivalence relation on the set of integers see Definition 1. Our next result says that we can add or subtract the same integer on both sides of a congruence, or multiply both sides of a congruence by the same integer.

This is an easy exercise using the definitions and also follows from Theorem 2. We cannot divide both sides of a congruence by the same integer, or cancel an integer from both sides, in general. The reason we cannot divide both sides of a congruence by an integer in general is that not every integer has a multiplicative inverse modulo n. As we noted when discussing the properties of our various number systems, division is actually multiplication by a multiplicative inverse.

Suppose a is an integer with a multiplicative inverse r modulo n. Suppose first that a has a multiplicative inverse r modulo n. For small values of a and n, an inverse of a can usually be found easily by trial and error.

This is stated more formally in the next theorem. Thus, even though the multiplicative inverse is not unique, all of the inverses must be congruent modulo n. It says that if the modulus is a prime, then the situation in Remark 2 above cannot happen. This set is a complete set of residues mod n by Theorem 2.

In modular arithmetic, we define addition and multiplication on N as follows. It can be shown that that N is a commutative ring with 1 under the operations of addition and multiplication modulo n.

The proof is rather tedious and somewhat tricky, however, due in part to the requirement that we reduce sums and products modulo n to get an element of N. This is similar to problems encountered in proving that the set Q of rational numbers is a field, arising from the existence of many equivalent expressions of a given fraction.

An advantage of this alternate construction is that the proof that it yields a ring is much more straightforward. Recall that Theorem 2. The set of equivalence classes will be the underlying set for our ring construction.

We must define the operations of addition and multiplication on the set Zn. Note, however, that there are many different representatives for a given congruence class see Proposition 1. The definitions of addition and multiplication seem to depend on the class representative. In this case, we get the same sum and product using either set of representatives.

In fact, this is true in any case. It follows from Theorem 2. Using the definitions of addition and multiplication in Zn and the fact that Z is a commutative ring with 1, it is straightforward to verify the next theorem.

There is an interesting difference between the rings Zn and Z. This is not the case in Zn. In particular, this implies Z6 cannot be a field. Hence, if n is not a prime, then Zn is not a field. This also follows from Theorem 2. An element a of Zn has a multiplicative inverse i. Now Zn is a field if and only if every non-zero element has a multiplicative inverse, so if and only if every integer not divisible by n is relatively prime to n.

This holds if and only if n is prime. We therefore have the following result. The ring Zn is a field if and only if n is a prime. This theorem gives us an infinite family of finite fields Zp , p a prime, in addition to the infinite fields Q, R, and C.

Show that every perfect square is congruent to 0, 1, or 4 modulo 8. Let n be a positive integer, a an integer, and let a be the congruence class of a modulo n. Show that Zn has precisely n elements. Determine which elements of Z10 have multiplicative inverses, and find the inverse of each. We will refer to dk as the first digit of m and d0 as the last digit of m.

There are nice shortcuts for finding the least residue of an integer modulo some small integers, and therefore for determining when an integer is divisible by these small integers. The corollaries below give tests for divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and Congruence and Divisibility by Powers of 2, 5, and 10 We first consider congruence and divisibility by powers of 2, 5, and The idea behind the proofs of the tests is demonstrated in the following examples.

Therefore, the least non-negative residue of m modulo 4 is 3. Therefore, the least non-negative residue of m modulo is Then Z m if and only if Z divides the number made up of the last n digits of m. Thus 2 m if and only if d0 is divisible by 2, i.

Since 0 and 5 are the only one-digit numbers divisible by 5, and 0 is the only one-digit number divisible by 10, statements ii and iii hold. Divisibility tests for 4 and 8 follow from Corollary 2. Since 4 - 54, we have that 4 - Since 8 check!

Since check! In other words, 3 divides m if and only if 3 divides the sum of the digits of m, and 9 divides m if and only if 9 divides the sum of the digits of m. In particular, 3 - and 9 - In particular, 3 30, so 3 , but 9 - Hence 6 m if and only if m passes the divisibility tests for both 2 and 3, and 12 m if and only if m passes the divisibility tests for both 4 and 3. We will require the following notation.

Note that the signs alternate beginning with a plus for d0 and a minus for d1. When computing A to test for congruence modulo 11, it is essential to add the digits in the places corresponding to even powers of 10 and subtract those in the places corresponding to odd powers of In particular, 11 - Hence, by Corollary 2. As with the computation of the alternating sum A, it is essential to compute T using the correct signs in order to apply the tests for congruence modulo 7 and This test is much less convenient than the tests for 11 given in Theorem 2.

In particular, 7 - 23, , , Use the divisibility tests for 2 and 7 to derive a test for divisibility by Prove that the test is valid. Use the divisibility tests for 3 and 5 to derive a test for divisibility by Use the divisibility tests for 2 and 9 to derive a test for divisibility by On Exercises 9—13, determine which, if any, of the given integers a, b, c are divisible by the indicated integer n.

Show your work and justify your answers. We can define addition and multiplication of polynomials, and we will see that the set of polynomials has algebraic properties very similar to those of the integers.

We will consider algebraic properties such as those in Definition 1. Unless otherwise stated, all definitions and results are valid in all five cases. Definition 3. The set of all polynomials in x with coefficients in S is denoted S[x]. We will need the following basic terminology.

The elements ai are called the coefficients of p x. The monomials ai xi are the terms of p x. The term with the highest power of x and non-zero coefficient is the leading term of p x , and a0 is the constant term.

The leading coefficient of p x is the coefficient of the leading term of p x. We call p x a monic polynomial if the leading coefficient of p x is 1. If ai is non-zero, we call i the degree of the monomial ai xi. If p x is a non-zero polynomial, the degree of the polynomial p x is defined to be the degree of the leading term of p x.

We denote the degree of p x by deg p x. A polynomial of degree 1 is called a linear polynomial, a polynomial of degree 2 is a quadratic polynomial, and a polynomial of degree 3 is a cubic polynomial. Note that this is the case regardless of the order in which the terms are written. The leading term is the term with the highest power of x.

Another approach used by some is to simply say that the zero polynomial has no degree. In any case, the degree of the zero polynomial cannot be defined to be 0 or any other integer. Otherwise, certain desirable properties of the degree will not be valid. Also, according to our definition, the zero polynomial has no leading term or leading coefficient.

In many ways, the set S[x] is algebraically very similar to the set Z of integers. In particular, we can define addition and multiplication of polynomials and show that these operations satisfy the same basic algebraic properties in S[x] as they do in Z. To compare or add two polynomials of different degrees, we can write both using all powers of x up to the higher degree, using 0 as a coefficient when necessary.

Thus we add polynomials by simply adding corresponding coefficients. This example is intended to illustrate the notation in the formal definition of multiplication. In practice, polynomials are multiplied by multiplying every term of the first by every term of the second using laws of exponents and multiplying coefficients , and then collecting like terms.

Theorem 3. Property Relating Addition and Multiplication: x. Before considering the existence of multiplicative inverses, we prove some useful properties of the degrees of polynomials.

We may now assume p x and q x are both non-zero. Thus, by Definition 3. This would not be true had we attempted to define the degree of the zero polynomial to be 0 or any other integer.

Corollary 3. A polynomial p x in S[x] has a multiplicative inverse in S[x] if and only if p x is a non-zero constant polynomial. Hence deg p x and deg q x are both non-negative integers and their sum is 0. We can say the following, however.

By Definition 3. By Corollary 3. Hence S[x] is never a field. However, it is possible to construct a field containing S[x] in much the same way we constructed the field Q containing the integers Z. It is then straightforward, but tedious, to verify that R is a field. Identifying a polynomial p x in S[x] with the equivalence class [ p x , 1 ] in R, we consider S[x] to be be contained in the field R. We then obtain the more familiar field of rational functions, denoted S x.

By identifying a polyno- p x mial p x with the rational function , we consider S[x] to be contained in S x , just as Z is 1 contained in Q. Use only Definition 3. Include any 0 coefficients as well. Use Definition 3. That is, verify the commutative law of polynomial addition. Be sure to indicate where this is used. That is, verify the associative law of polynomial addition. That is, verify that R[x] is closed under multiplication.

If we were to expand this product, we would obtain a polynomial in x of degree n with integer coefficients. The coefficients are the binomial coefficients. Hence the n coefficient r is the number of ways to choose r factors from the set of n factors.

Proposition 3. Those in between can then be calculated inductively using the following very important theorem. Recall that for a positive integer m, we define m-factorial, denoted m!

We proceed by induction on n. Using Proposition 3. The second equality in the formula follows by replacing the binomial coefficient by the expression from Theorem 3. Evaluate the following binomial coefficients. Show your work. Show that if n and r are integers with 0 6 r 6 n, then r! Most of the results on divisibility of integers can be translated almost word for word into analogous results for polynomials.

Note: So that we can always divide coefficients of polynomials, we will now assume that the ring S of coefficients is one of Q, R, C, or Zp , p a prime, but not Z. Thus we assume that S is a field. In all of the results below, a x , b x , c x , etc. Recall that constants elements of S are considered to be elements of S[x] by identifying them with constant polynomials. As with integers, if a x b x , we also say a x is a divisor or factor of b x , that b x is a multiple of a x , or that b x is divisible by a x.

The following basic properties are proved in exactly the same way as the analogous results for integers. If a x is any polynomial, then 1 a x. The proofs of iv and v are left as exercises see Exercises 3. The analogous result for polynomials involves constant multiples. This is also necessary for the following corollary. The degree induces a partial order on the set of polynomials, and we obtain the following analogous result.

By Theorem 3. The analogue for polynomials is the following result. In this case, the process continues until the remainder has lower degree than the divisor. As before, the quotient and remainder are unique. The procedure is similar to long division of integers and is as follows: 1. Divide the leading term of the dividend by the leading term of the divisor.

Multiply the divisor by the result of Step 1. Subtract the result of Step 2 from the dividend. Using the result of Step 3 as the new dividend of lower degree , repeat the steps until the difference obtained in Step 3 is of strictly lower degree than the degree of the divisor. The sum of the terms obtained from Step 1 will be the quotient q x and the final remainder in Step 4 will be the remainder r x. The complete procedure for our example is given below. Again, the proof involves the long division procedure along with induction, and is omitted here.

As with integers, b x is the dividend, a x is the divisor, q x is the quotient, and r x is the remainder. Greatest Common Divisor Definition 3. The greatest common divisor of a x and b x is the monic polynomial d x satisfying i. The GCD of a x and b x is the monic polynomial of highest degree that divides both a x and b x.

It is not difficult to see that a GCD must exist, but it is not obvious from the definition that it must be unique. The procedure is nearly identical to that used for integers.