Mathematics BSc
Course description
2013. Algebra 4
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
2 + 2 2 + 3 exam +
pure math. mm1c1al4m
mm1c2al4m
4 compulsory
Strong Weak Prerequisites
Practice class
Strong:
Algebra3L-p (mm1c1al3m)
Lecture
Weak:
practice class
Prerequisites
Classical and linear algebra, elementary number theory, group- and ring theory.
Literature
• D. S. Dummit, R. M. Foote: Abstract algebra. Wiley, 2004.
Syllabus
• Field extensions. The notion of a field extension, its degree, extension by given elements. Minimal polynomial, algebraic, resp. transcendental elements. Characterization of minimal polynomials using irreducibility. Simple extensions, description of simple algebraic, resp. transcendental extensions. The degree of the extension equals the degree of the minimal polynomial. Simple extensions as quotient rings. Construction of simple extensions. The degree of consecutive extensions is the product of the degrees. Consequences: the degree of an element divides the degree of the extension, every finite extension is algebraic. Estimation for the degree of a sum and a product. The algebraic elements form a subfield. The field of algebraic numbers; it is algebraically closed. The existence of an algebraically closed extension (without proof).
• Galois theory. Splitting field. Normal extension, the splitting field of a polynomial is normal. The uniqueness of the splitting field. Perfect fields, every field of characteristic zero is perfect. Finite extensions of perfect fields are simple extensions. Relative automorphisms, the Galois group. The fundamental theorem of Galois theory. Algebraic conjugates; these are the roots of the minimal polynomial.
• Finite fields. Existence and uniqueness. Finite fields are perfect. The multiplicative group of a finite field is cyclic. Finite extensions of finite fields are always normal, and have cyclic Galois group generated by the Frobenius automorphism. The number and degree of intermediate fields. Wedderburn’s Theorem: Every finite division ring is commutative.
• Constructions with straight-edge and compass. Connection between steps of geometric constructions and field extensions of degree 2. Characterization of constructible numbers: the minimal polynomial has splitting field of 2-power degree. Impossibility of some constructions: duplication of the cube, trisection of angles, squaring the circle. The degree and Galois group of the cyclotomic fields. Characterization of constructible regular polygons.
• Solving equations by radicals. Connection with solvability of the Galois group. Example for a polynomial with Galois group S5 that consequently cannot be solved by radicals. The Galois group of the general equation is the full symmetric group. There is no formula for solving equations of degree 5 or greater.
• Modules. Examples: Abelian groups, vector spaces, F[x]-modules. Submodules, homomorphisms, quotient modules, Homomorphism Theorem. Direct sums (with possibly infinitely many summands). Free modules. Simple (minimal) modules. Endomorphism ring of a module. Schur’s Lemma. Jacobson’s Density Theorem. Connection between idempotent endomorphisms and direct decompositions. Commutative diagrams. Exact sequences. Projective modules. A module is projective iff it is a direct summand of a free module. Injective modules. Over a commutative ring R, Hom(A,B) is also an R-module. Tensor product of modules.
• Finitely generated modules over principal ideal domains. Normal form of matrices. Cyclic modules, their decompositions. The Fundamental Theorem of finitely generated modules over principal ideal domains. Applications: finite Abelian groups, Jordan canonical form of matrices.
• Semisimple modules and rings. Equivalent characterizations of semisimple rings: every module is semisimple; every module is projective; every module is injective; left ideals of the ring satisfy the descending chain condition and no nilpotent left ideal is contained in the ring. Jacobson radical. Decomposition of semisimple modules into direct sum of homogeneous submodules. Wedderburn–Artin Theorem: decomposition of semisimple rings into direct sum of full matrix rings.
• Algebras over fields. Examples: ring of polynomials, matrix ring, field extension. Group algebras, Maschke’s Theorem. The algebra of quaternions. Frobenius’ Theorem on finite dimensional algebras without zero divisors over the real field.