The main aim of this course is to give to the student the basic properties and theorems in group theory and ring theory, and, in addition to familiarize the student with the arithmetic in ℤ and K [X].
Pedagogical Objectives: At the end of this course, the student should be able to:
1. Recognize the structure of a group on a set and give examples of groups
2. Find the table of the laws of certain finite groups..
3. Know and use the isomorphism theorems of groups.
4. Find the kernel and the image of a homomorphism de groups.
5. Prove that a subgroup of a group is normal and construct the corresponding quotient group.
6. Determine the order of an element in a finite group.
7. Know and use the properties of a cyclic group.
8. Give examples in rings and fields.
9. Construct isomorphisms of rings.
10. Know how to prove that a non empty subset with some properties is a left or right or a 2-sided ideal of a ring.
11. Construct the quotient ring.
12. Know and use the isomorphism theorems in a ring .
13. State and use the theorem of the Euclidean division in ℤ.
14. Define and use the properties of G.C.D. and L.C.M of a finite family of ℤ .
15. Know the prime numbers and the numbers which are relatively prime .
16. State the fundamental theorem of an integer as a product of a finite prime integers.
17. Know to use the Euclid’s algorithm and Bezout’s theorem.
18. State Gauss’ theorem and know Bezout’s identity.
19. Know the residual ring and solve linear congruences in ℤ.
20. Find the Euclidean division of a polynomial in K[x] by another.
21. Determine the G.C.D. and the L.C.M. of wo polynomials .
22. Determine the irreducible polynomials of R[x] and D[x].
23. Factorize a polynomial of K[x] into product of irreducible polynomials (up to unit).
24. Know the criteria of irreducibilities of a polynomial of ℚ [X].
25. Know the relation between the roots and the coefficients of a polynomial.
1. Group theory: Subgroups. Homomorphisms. Normal subgroups. Subgroups of ℤ. Quotient group. Lagrange’s theorem. Isomorphism theorem. Group poduct. Order of an element. Cyclic group. Examples .
2. Ring Theory: Subrings. Homomorphisms.ideals. Quotient rings. Isomorphism theorem.Characteristic. Prime ideals. Integral domain. Principal ideal ring. Fields. Example:ℤ and ℤ/pℤ.
3. The integers : Principle of induction.The Euclidean property. Prime numbers. Relatively prime numbers. G.C.D . L.C.M . Linear congruence in ℤ. Chinese remainder theorem.
4. Polynomials: The division Algorithm. P. I .D . Arithmetic in F [X]. Irreducible polynomials. Root of a Polynomial. Polynomials in ℤ[X] and ℚ[X]. The Eisenstein Criterion.
1. It is necessary to take care that the student understands with facility “ finite Groups” of order 1 until order 6 and knows well the techniques allowing to find subgroups of a finite group.
2. It is necessary to insist on the major significance of isomorphic groups
3. The three isomorphism theorems : Imf ; and
4. It is necessary to multiply the examples in groups and rings.
5. We state Krull ’s theorem to prove that every proper ideal in a unitary ring is contained in a maximal ideal
In arithmetic: The proof of the Euclidean division theorem is given. The student will learn how to calculate the G.C.D. and the L.C.M of two elements of ℤ by different methods . The student will learn how to express the G.C.D. of 2 elements a and b as
( d = ua + gb ).
We prove that the set of all prime numbers in ℤ is infinite .
The student learn when a polynomial is irreducible in ℚ [X] Using the Eisenstein Criterion.