Abstract Algebra: Groups, Rings & Fields
Video Lectures
Displaying all 83 video lectures.
Lecture 1 Play Video |
Mystery Division Problem Abstract Algebra: A pattern is given for a long division problem. That is, only the lengths of digits used are given. We determine the divisor, dividend, and quotient. |
I. Group Theory | |
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Lecture 2 Play Video |
GT1. Definition of Group Abstract Algebra: We introduce the notion of a group and describe basic properties. Examples given include familiar abelian groups and the symmetric groups. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 3 Play Video |
GT1.1. Example of Group Inverse Abstract Algebra: We note some useful rules for working with inverses in groups. 1) It is enough to find an inverse on one side only, 2) inverses are unique, and 3) the inverse of a product is the reversed product of the inverses. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 4 Play Video |
Order 2 Elements in Finite Group Abstract Algebra: Let G be a finite group. (1) If |G| is even, show that G has an odd number of elements of order 2. (2) If G is abelian, we compute the sum of the elements of the group (where group multiplication is written as addition). |
Lecture 5 Play Video |
Example of Group Cancellation Law Abstract Algebra: Let G be a nonempty finite set with multiplication. Suppose the multiplication is associative and satisfies cancellation laws on both sides. Show that G is a group. |
Lecture 6 Play Video |
GT2. Definition of Subgroup Abstract Algebra: We define the notion of a subgroup and provide various examples. We also consider cyclic subgroups and subgroups generated by subsets in a given group G. Example include A4 and D8. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 7 Play Video |
Example of Group: GL(2, R) (1 of 3) Abstract Algebra: Let G = GL(2,R) be the set of real 2x2 invertible matrices. In this first part, we show that G is a group. Using the identity det(AB)=det(A)det(B), we give an indication of how to extend to nxn invertible matrices. |
Lecture 8 Play Video |
GT3. Cosets and Lagrange's Theorem Abstract Algebra: Let G be a group with subgroup H. We define an equivalence relation on G that partitions G into left cosets. We use this partition to prove Lagrange's Theorem and its corollary. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 9 Play Video |
GT4. Normal Subgroups and Quotient Groups Abstract Algebra: We define normal subgroups and show that, in this case, the space of cosets carries a group structure, the quotient group. Example include S3, the modular integers, and Q/Z. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 10 Play Video |
GT5. Index 2 Theorem and Dihedral Groups Abstract Algebra: We state and prove the Index Two Theorem for finding normal subgroup and list several examples. These include S3, A4, and the symmetry groups for the regular n-gon, D_2n. We give several presentations of the latter groups and calculate the center. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 11 Play Video |
Example of Group: GL(2,R) (2 of 3) Abstract Algebra: Let G = GL(2, R) be the group of real 2 x2 invertible matrices, and let H be the subset of matrices with determinant = 1. We show that H is a normal subgroup of G directly and by exhibiting H as the kernel of a homomorphism. |
Lecture 12 Play Video |
Example of Group: GL(2, R) (3 of 3) Abstract Algebra: Let G=GL(2, R) be the group of real invertible 2x2 matrices. We consider two group actions for the group GL(2, R) on itself. We interpret the results in terms of linear algebra and change of basis. We also explain how conjugacy classes of G relate to the diagonalization procedure. |
Lecture 13 Play Video |
GT6. Centralizers, Normalizers, and Direct Products Abstract Algebra: We consider further methods of constructing new groups from old. We consider centralizer and normalizer subgroups, which are useful when the group is non-abelian, and direct products. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 14 Play Video |
GT7. The Commutator Subgroup Abstract Algebra: We define the commutator subgroup for a group G and the corresponding quotient group, the abelianization of G. The main example is the dihedral group, which splits into two cases. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 15 Play Video |
GT8. Group Homomorphisms Abstract Algebra: We define homomorphism between groups and draw connections to normal subgroups and quotient groups. Precisely the kernel of a homomorphism is a normal subgroup, and we can associate a surjective homomorphism to every normal subgroup. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 16 Play Video |
GT9. Group Isomorphisms Abstract Algebra: In analogy with bijections for sets, we define isomorphisms for groups. We note various properties of group isomorphisms and a method for constructing isomorphisms from onto homomorphisms. We also show that isomorphism is an equivalence relation on the class of groups. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 17 Play Video |
Example of Group Isomorphism Abstract Algebra: An abelian group G has order p^2, where p is a prime number. Show that G is isomorphic to either a cyclic group of order p^2 or a product of cyclic groups of order p. We emphasize that the isomorphic property usually requires construction of an isomorphism. |
Lecture 18 Play Video |
GT10. Examples of Non-Isomorphic Groups Abstract Algebra: We note some items that distinguish non-isomorphic groups, including cardinality, orders of elements and subgroups, abelian and cyclic properties, and normality of subgroups. We give a classification of all groups with order less than or equal to 7. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 19 Play Video |
GT11. Group Automorphisms Abstract Algebra: We consider the group Aut(G) of automorphisms of G, the isomorphisms from G to itself. We show that the inner automorphisms of G, induced by conjugation, form a normal subgroup Inn(G) of Aut(G), and that Inn(G) is isomorphic to G/Z(G). Examples include Z/3, Z/4, Z, Z/2 x Z/2, and S3. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 20 Play Video |
GT11.1. Automorphisms of A4 Abstract Algebra: We compute the automorphism group of A4, the alternating group on 4 letters. We have that Aut(G) = S4, the symmetric group on 4 letters, Inn(A4) = A4, and Out(A4)=Z/2. We note that the coset structure splits S4 into even and odd permutations. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 21 Play Video |
GT12. Aut(Z/n) and Fermat's Little Theorem Abstract Algebra: We show that Aut(Z/n) is isomorphic to (Z/n)*, the group of units in Z/n. In turn, we show that the units consist of all m in Z/n with gcd(m,n)=1. Using (Z/n)*, we define the Euler totient function and state and prove Fermat's Little Theorem: if p is a prime, then, for all integers k, p divides k^p - k. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 22 Play Video |
GT12.1. Automorphisms of Dihedral Groups Abstract Algebra: We compute Aut(G), Inn(G), and Out(G) when G is a dihedral group D_2n. We also show that Aut(D_2n) always contains a subgroup isomorphic to D_2n and that Aut(D_2n) may be realized as a matrix group with entries n Z/n. |
Lecture 23 Play Video |
GT13. Groups of Order 8 Abstract Algebra: We classify all groups of order 8 up to isomorphism. There are 3 abelian isomorphism classes and two non-abelian classes, the symmetry group of the square D8 and the quaternion group Q. We define and describe Q and compute Inn(Q) and Out(Q). U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 24 Play Video |
GT14. Semidirect Products Abstract Algebra: Using automorphisms, we define the semidirect product of two groups. We prove the group property and construct various examples, including the dihedral groups. As an application, we show that there exist only two isomorphism classes for groups of order 10 (actually 2p with p an odd prime). U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 25 Play Video |
GT15. Group Actions Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetries of a set X. A given group action defines an equivalence relation, which in turn yields a partition of X into orbits. Orbits are also described as cosets of the group. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 26 Play Video |
GT16. Cayley's Theorem Abstract Algebra: We consider the left regular action of G on the set X = G. We prove Cayley's Theorem, that every group is isomorphic to a subgroup of a symmetric group, and note a variant when X=G/H. As an application, we show that every group of order p^2 with p prime is abelian. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 27 Play Video |
GT16.1 Examples of Cayley's Theorem Abstract Algebra: We give further examples of Cayley's Theorem and its variant. Examples include the real numbers, the symmetry group of the square, and the quaternion group. |
Lecture 28 Play Video |
GT17. Symmetric and Alternating Groups Abstract Algebra: We review symmetric and alternating groups. We show that S_n is generated by its 2-cycles and that A_n is generated by its 3-cycles. Applying the latter with the Conjugation Formula, we show that A_5 is simple. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 29 Play Video |
Order 12 Subgroups in S5 Abstract Algebra: Find all subgroups in S5, the symmetric group on 5 letters, that are isomorphic to D12, the dihedral group with 12 elements. |
Lecture 30 Play Video |
GT17.1. Permutation Matrices Abstract Algebra: (Linear Algebra Required) The symmetric group S_n is realized as a matrix group using permutation matrices. That is, S_n is shown to the isomorphic to a subgroup of O(n), the group of nxn real orthogonal matrices. Applying Cayley's Theorem, we show that every finite group is isomorphic to a subgroup of O(n). U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 31 Play Video |
|Z(G)| for |G|=pq Abstract Algebra: Let G be a group of order pq, where p and q are distinct primes. We show that either G is abelian or Z(G) = {e}. We give two proofs: the first uses the class equation, the second uses more elementary methods. |
Lecture 32 Play Video |
GT18. Conjugacy and The Class Equation Abstract Algebra: We consider the group action of the group G on itself given by conjugation. The orbits, called conjugacy classes, partition the group, and we have the Class Equation when G is finite. We also show that the partition applies to normal subgroups. Finally we apply the class equation to groups of prime power order. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 33 Play Video |
Class Equation for Dihedral Group D8 Abstract Algebra: Consider the dihedral group with eight elements D8, the symmetries of the square. Find all conjugacy classes of D8, and verify the class equation. Then find all subgroups and determine which ones are normal. |
Lecture 34 Play Video |
GT18.1. Class Equation for Dihedral Groups Abstract Algebra: We consider the class equation for the dihedral groups D_2n. Conjugacy classes are computed, and we verify the cardinality equation using centralizers. To finish, we consider the partitions for normal subgroups. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 35 Play Video |
GT18.2. A_n is Simple (n ge 5) Abstract Algebra: Using conjugacy classes, we give a second proof that A5, the alternating group on 5 letters, is simple. We adapt the first proof that A5 is simple to show that An is simple when n is greater than 5. The key step is to show that any normal subgroup with more than the identity contains a 3-cycle. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 36 Play Video |
GT19. Cauchy's Theorem Abstract Algebra: Cauchy's Theorem states that if p is a prime that divides the order of a finite group G, then there exists an element of order p. We give a proof using the class equation and, as an application, we show that the isomorphism class for a group of order 15 is unique. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 37 Play Video |
GT20. Overview of Sylow Theory Abstract Algebra: As an analogue of Cauchy's Theorem for subgroups, we state the three Sylow Theorems for finite groups. Examples include S3 and A4. We also note the analogue to Sylow Theory for p-groups. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 38 Play Video |
GT20.1. Sylow Theorems - Proofs Abstract Algebra: We give proofs of the three Sylow Theorems. Techniques include the class equation and group actions on subgroups. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 39 Play Video |
GT20.2 Sylow Theory for Simple 60 Abstract Algebra: Using Sylow theory, we show that any simple, non-abelian group with 60 elements is isomorphic to A_5, the alternating group on 5 letters. As an application, we show that A_5 is isomorphic to the symmetry group of rigid motions of a regular icosahedron. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 40 Play Video |
Sylow Theory for Order 12 Groups 1 Abstract Algebra: Let G be a finite group of order 12. We apply Sylow theory to study such groups. In Part 1, we consider the abelian cases and A4, the alternating group on 4 letters. |
Lecture 41 Play Video |
Sylow Theory for Order 12 Groups 2 Abstract Algebra: Let G be a finite group of order 12. Using Sylow Theory, we consider the isomorphism types of G when n_3 = 1 and n_1. In this case, G is isomorphic to either D_12, the symmetry group of a regular hexagon, or a nontrivial semidirect product of Z/3 and Z/4. |
Lecture 42 Play Video |
Simple Group 168 - Sylow Theory - Part 1 Abstract Algebra: Let G be a simple group of order 168. We calculate the number of Sylow subgroups, number of elements of a given order, and conjugacy class structure. In Part 1, we consider Sylow-p subgroup for p = 3, 7. |
Lecture 43 Play Video |
Simple Group 168 - Sylow Theory - Part 2 Abstract Algebra: Let G be a simple group of order 168. In Part 2, we compute the number of Sylow-2 subgroups and show that each Sylow-2 subgroup is isomorphic to D_8, the symmetry group of the square. |
Lecture 44 Play Video |
GT21. Internal Products Abstract Algebra: We consider conditions for when a group is isomorphic to a direct or semidirect product. Examples include groups of order 45, 21, and cyclic groups Z/mn, where m,n are relatively prime. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 45 Play Video |
GT22. The Fundamental Theorem of Finite Abelian Groups Abstract Algebra: We state and prove the Fundamental Theorem of Finite Abelian Groups. We apply internal direct products to Sylow subgroups in this case. Steps include showing the result for finite abelian p-groups and using the combination rule for cyclic groups with relatively prime order. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html |
Lecture 46 Play Video |
GT23. Composition and Classification Abstract Algebra: We use composition series as another technique for studying finite groups, which leads to the notion of solvable groups and puts the focus on simple groups. From there, we survey the classification of finite simple groups and the Monster group. |
II. Ring Theory | |
Lecture 47 Play Video |
RNT1.1. Definition of Ring Ring Theory: We define rings and give many examples. Items under consideration include commutativity and multiplicative inverses. Example include modular integers, square matrices, polynomial rings, quaternions, and adjoins of algebraic and transcendental numbers. |
Lecture 48 Play Video |
RNT1.2. Definition of Integral Domain Ring Theory: We consider integral domains, which are commutative rings that contain no zero divisors. We show that this property is equivalent to a cancellation law for the ring. Finally we note some basic connections between integral domains and fields. |
Lecture 49 Play Video |
RNT1.2.1. Example of Integral Domain Abstract Algebra: Let R be an integral domain. Suppose there exists a y in R such that y + ... + y (n times) = 0. Show that x + ... + x (n times) = 0 for all x in R. |
Lecture 50 Play Video |
RNT1.2.2. Order of a Finite Field Abstract Algebra: Let F be a finite field. Prove that F has p^m elements, where p is prime and m gt 0. We note two approaches: one uses the Fundamental Theorem of Finite Abelian Groups, while the other uses linear algebra. |
Lecture 51 Play Video |
RNT1.3. Ring Homomorphisms Ring Theory: We define ring homomorphisms, ring isomorphisms, and kernels. These will be used to draw an analogue to the connections in group theory between group homomorphisms, normal subgroups, and quotient groups. |
Lecture 52 Play Video |
RNT1.4. Ideals and Quotient Rings Ring Theory: We define ideals in rings as an analogue of normal subgroups in group theory. We give a correspondence between (two-sided) ideals and kernels of homomorphisms using quotient rings. We also state the First Isomorphism Theorem for Rings and give examples. |
Lecture 53 Play Video |
RNT1.4.1. Example of Quotient Ring Abstract Algebra: Are there fields F such that the rings F[x]/(x^2) and F[x]/(x^2-1) are isomorphic? We construct an isomorphism when char F = 2. |
Lecture 54 Play Video |
RNT2.1. Maximal Ideals and Fields Ring Theory: We now consider special types of rings. In this part, we define maximal ideals and explore their relation to fields. In addition, we note three ways to construct fields. |
Lecture 55 Play Video |
RNT2.1.1. Finite Fields of Orders 4 and 8 Ring Theory: As an application of maximal ideals and residue fields, we give explicit constructions of fields with 4 and 8 elements. A key step is to find irreducible polynomials (quadratic and cubic). |
Lecture 56 Play Video |
RNT2.2. Principal Ideal Domains Ring Theory: We define PIDs and UFDs and describe their relationship. Prime and irreducible elements are defined, and conditions for implication are given. (second version: corrections to definition of prime and irreducible; comment should be 'R UFD implies R[x] UFD, improved proof that, in UFD, irreducible implies prime). |
Lecture 57 Play Video |
RNT2.3. Euclidean Domains Ring Theory: We define Euclidean domains as integral domains with a division algorithm. We show that euclidean domains are PIDs and UFDs, and that Euclidean domains allow for the Euclidean algorithm and Bezout's Identity. |
Lecture 58 Play Video |
RNT2.3.1. Euclidean Algorithm for Gaussian Integers Ring Theory: We use the Euclidean algorithm to find the GCD of the Gaussian integers 11+16i and 10+11i. Then we solve for the coefficients in Bezout's identity in this case. |
Lecture 59 Play Video |
RNT2.4. Gaussian Primes Ring Theory: As an application of all previous ideas on rings, we determine the primes in the Euclidean domain of Gaussian integers Z[i]. Not only is the answer somewhat elegant, but it contains a beautiful theorem on prime integers due to Fermat. We finish with examples of factorizations in Z[i]. |
Lecture 60 Play Video |
RNT2.5. Polynomial Rings over Fields Ring Theory: We show that polynomial rings over fields are Euclidean domains and explore factorization and extension fields using irreducible polynomials. As an application, we show that the units of a finite field form a cyclic group under multiplication. |
Lecture 61 Play Video |
RNT2.5.1. Euclidean Algorithm for Z/3[x] Ring Theory: Let f(x)=x^5+2x^3+2x^2 + x+2 and g(x)=x^4+2x^3+2x^2 be polynomials over Z/3. Use the Euclidean algorithm to find gcd(f,g), find the prime factorizations of f and g, and find coefficients for Bezout's Identity in this case. We also find a field in which f(x) factors into linear factors. |
Lecture 62 Play Video |
RNT2.6.1. Gauss' Lemma Ring Theory: We consider general polynomial rings over an integral domain. In this part, we show that polynomial rings over integral domains are integral domains, and we prove Gauss' Lemma as a step in showing that polynomial rings over UFDs are UFDs. |
Lecture 63 Play Video |
RNT2.6.2. Eisenstein's Criterion Ring Theory: Continuing with Gauss' Lemma, we prove Eisenstein's Criterion for Irreducibility and that R UFD implies R[x] UFD. As an example of EC, we show that f(x) = x^4+x^3+x^2+x+1 is irreducible over the integers using substitution. |
III. Field Theory | |
Lecture 64 Play Video |
FIT1.1. Number Fields Field Theory: We give a brief review of some of the main results on fields in basic ring theory and give examples to motivate field theory. Examples include field automorphisms for the rational polynomials x^2-2 and x^3-2. |
Lecture 65 Play Video |
FIT1.2. Characteristic p Field theory: We consider basic properties of finite fields and characteristic p. As in the previous part, we give motivation for Galois theory by considering automorphisms and fixed fields. |
Lecture 66 Play Video |
FIT2.1. Field Extensions Field Theory: Let F be a subfield of the field K. We consider K as a vector space over F and define the degree of K over F as the dimension. We give a degree formula for successive extensions, and consider extensions in terms of bases. |
Lecture 67 Play Video |
FIT2.2. Simple Extensions Field Theory: We consider the case of simple extensions, where we adjoin a single element to a given field. The cases of transcendental and algebraic arise, depending on whether the kernel of the evaluation map is zero or not. In the algebraic case, we define the minimal polynomial, show it is irreducible, and calculate the degree of the extension. |
Lecture 68 Play Video |
FIT2.2.1. Example: Cubic Extension Field Theory: Let a = 3 + 2^1/3 + 2 * 2^2/3. We show that 1) Q[a] = Q[2^1/3], 2) the minimal polynomial of a over Q is x^3 - 9x^2 + 15x - 25, and 3) a^-1 expressed as a polynomial in a over the rationals. The main technique from linear algebra is the minimal polynomial of a linear transformation. |
Lecture 69 Play Video |
FIT2.2.2. Example: Quartic Extension Field Theory: Let a = sqrt(2) +sqrt(3) in the reals. We show that 1) Q[a]=Q[sqrt(2),sqrt(3)], 2) the minimal polynomial over Q is x^4 - 10x^2 + 1, and 3) an inverse for a^-1 as a polynomial in a over Q. |
Lecture 70 Play Video |
FIT2.3.1. Algebraic Numbers Field Theory: We consider the property of algebraic in terms of finite degree, and we define algebraic numbers as those complex numbers that are algebraic over the rationals. Then we give an overview of algebraic numbers with examples. |
Lecture 71 Play Video |
FIT2.3.2. Cardinality and Transcendentals Field Theory: We show that the set of algebraic numbers is countable and that any extension of a countable field F by a transcendental is countable. We then give an overview of known results on transcendental numbers. |
Lecture 72 Play Video |
FIT2.3.3. Algebraic Extensions Field Theory: We define an algebraic extension of a field F and show that successive algebraic extensions are also algebraic. This gives a useful criterion for checking algberaic elements. We finish with algebraic closures. |
Lecture 73 Play Video |
FIT3.1.1. Roots of Polynomials Field Theory: We recall basic factoring results for polynomials from Ring Theory and give a definition of a splitting field. This allows one to consider any irreducible polynomial as a set of roots, and in turn we consider when an irreducible polynomial can have multiple roots. We finish with a definition of separable elements and extensions. |
Lecture 74 Play Video |
FIT3.1.2. Roots of Real Polynomials Field Theory: We now consider roots of real and complex polynomials. We state and prove the Fundamental Theorem of Algebra, and note its consequences for real polynomials. Then we consider the relation between splitting fields, automorphisms, and roots. |
Lecture 75 Play Video |
FIT3.1.3. Example of Splitting Field Field Theory: Let f(x) = x^4 -16x^2 +4. We find the roots of f(x), calculate the splitting field K of f(x) over Q in C, and determine the automorphism group of K. |
Lecture 76 Play Video |
FIT3.1.4. Factoring Example: Artin-Schreier Polynomials Field Theory: We show that g(x)=x^5-x+1 is irreducible over the rationals using techniques from finite fields. This leads to the definition of an Artin-Schreier polynomial, and in turn we obtain a class of irreducible polynomials over the rationals and prime characteristic. |
Lecture 77 Play Video |
FIT3.2.1. Cyclotomic Polynomials Field Theory: We define cyclotomic polynomial as the minimal polynomials of roots of unity over the rationals. We show that the roots of the N-the cyclotomic polynomial are precisely the primitive N-th roots of unity, that the coefficients are integers, and that the degree is phi(N), where phi is the Euler totient function. We also note another formula for cyclotomic polynomials using Mobius inversion. |
Lecture 78 Play Video |
FIT3.2.2. Mobius Inversion Formula Field Theory/Number Theory: We state and prove the Mobius Inversion Formula. We apply the formula to several examples, including cyclotomic polynomials and the Euler totient function. (Reload: Badly synced audio after compression.) |
Lecture 79 Play Video |
FIT4.1. Galois Group of a Polynomial Field Theory: We define the Galois group of a polynomial g(x) as the group of automorphisms of the splitting field K that fix the base field F pointwise. The Galois group acts faithfully on the set of roots of g(x) and is isomorphic to a subgroup of a symmetric group. We also show that this action is transitive when g(x) is irreducible over F. |
Lecture 80 Play Video |
FIT4.2. Automorphisms and Degree Field Theory: Having established an estimate on the size of a Galois group of a polynomial, we look at general automorphism groups of finite extensions and give an upper bound on group order by the degree of the extension That is, when [K:F] finite, |Aut(K/F)| is less than or equal to [K:F]. |
Lecture 81 Play Video |
FIT4.3. Galois Correspondence 1 - Examples Field Theory: We define Galois extensions and state the Fundamental Theorem of Galois Theory. Proofs are given in the next part; we give examples to illustrate the main ideas. |
Lecture 82 Play Video |
FIT4.3.1. Galois Group of Order 8 Field Theory: Let K be Q[sqrt(2), sqrt(3), sqrt(5)], the splitting field of f(x) = (x^2-2)(x^2-3)(x^2-5) over Q. Find the Galois group of K over Q, find all subgroups of the Galois group, and find all subfields of K over Q. |
Lecture 83 Play Video |
FIT4.3.2. Example of Galois Group over Finite Field Field Theory: We compare the splitting fields of the polynomial f(x)=x^8-1 over the rationals and Z/5. We compute the Galois groups and identify Galois correspondences. |