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What's new!
o Jan. 8: We have covered most of Sections 18 and 19. We will talk about homomorphism and isomorphism of rings on Friday. o Jan. 13: Today we will give examples of homomorphisms, then will start Sec. 21 (The field of quotients of an integral domain). o Jan. 15: Briefly discussed Sec. 20 (Fermat and Euler's theorems). We finished Sec. 21 (Field of quotients). Next time Sec. 22 (Rings of polynomials). o Jan 23: On Friday we finish Sec. 23. We will briefly discuss Sec. 24 (only quaternions). We will skip Sec. 25 and will start with ideals (Sec. 26). o Jan 24: We discussed that irreducibility of a polynomial depends on the field: a polynomial which is irreducible over a field F might become reducible over a bigger field E. We used Eisenstein criterion to prove that cyclotomic polynomial x^p-1 + x^p-2 ... + 1 (where p is a prime) is irreducible over Q. We recalled that every polynomial over R factors completely over C (Fundamental Theorem of Algebra). We discussed what the complex roots of x^n - 1 are (n-th roots of unity). o Jan. 27: We proved Eisenstein criterion (Sec. 23). o Jan. 29: We discussed cyclotomic polynomial for n not necessarily prime. When n is not prime, the polynomial x^n-1 + x^n-2 + ... + 1 is not irreducible. The cyclotomic polynomial is an irreducible factor of this polynomial. It is the smallest polynomial (with coefficients in Z) which has the n-th root of unity e^(2*pi*i/n) as a root. The (complex) numbers e^(2*pi*i*k/n) when gcd(k, n) = 1 are called "primitive n-th roots of unity". o Jan. 29: We introduced the notions of a group ring, and quaternions (Sec. 24). o Jan. 31: We discuss ideals and homomorphisms (Sec. 26). Next week we cover Sec. 27 and if there is time Sec. 29. For now we will skip Sec. 28, we may come back to it later. o Feb. 10: We have covered homomorphisms and factor rings, maximal ideals, prime ideals and principal ideals. Next time we will prove that every ideal in Z or F[x] is principal. o Feb 15: Wikipedia page of Sir William Rown Hamilton o Feb. 15: Wikipedia page for cyclotomic polynomials . (Just for extra reading if interested.) o Solution to Problem 1 in HW4 corrected. x^3+2x^2+2x+1 is reducible over Q and factors as (x^2+x+1)(x+1). o Feb. 17: A fun math link for Valentine's Day! o Feb. 21: We almost finished Sec. 31 (algebraic extensions). We assumed everybody knows linear algebra and skipped Sec. 30 (Vector Spaces). You need to know the notion of "basis" from linea algebra. Next time we will discuss the notion of "algebraic closure". o Feb. 23: Canada won gold medal in hockey! Unfortunately US lost to Finland :( o By the way, our TA's MAC hours are: Thursdays 10-11 and Fridays 11-12 (you can find it at the bottom of the page too). o A funny clip for the occasion of winter oplympics. o March 3: Wikipedia page for constructible polygons . (Just for extra reading if interested.) o March 3: Wikipedia page for angle trisection . (Just for extra reading.) o March 3: Wikipedia page for squaring the circle . (Just for extra reading.) o HW7 posted as well as solutions to midterm and previous homeworks. o HW8 posted. o March 24: After finishing "Geometric Constructions" (Sec. 32) we covered "Finite Fields" (Sec. 33). o March 26: We have started with Chapter X (Galois Theory). We covered Sec. 48. We are discussing Sec. 49. Today we proved the "Isomorphism Extension Theorem". We will talk about its corollaries next time. o March 26: HW9 posted tonight. It is due Wednesday April 2nd. o March 30: Solutions to HW7 and HW8 posted. o April 3: HW10 posted. o April 4:We have finished Section 50 "Splitting sields". Splitting fields of polynomials are extensions E/F for which we have |G(E/F)| = {E:F}. Today we start Section 51 "Separable extensions". An extension E over F is separable if no element in E is a repeated root of its irreducible polynomial. It turns out that the almost always extensions are separable. We will also prove "Primitive element theorem" which states that: every finite separable extension is simple. We will use the fact that a is a repeated root of a polynomial f if f(a) = 0 and f'(a) = 0. This we discussed before in class. In Fraleigh this is discussed in Ex. 15-22 of Section 51. o Wikipedia page of Primitive Element Theorem . (Just for extra reading.) o Last HW posted. It is due on Wednesday April 16. o Wikipedia page of Evariste Galois . o A page in Science4all about Galois . o A page of Galois' writtings ! o April 13: Info for final exam posted. o April 13: Final exam is on Monday April 21 at 4pm at G37 Benedum (same location as the lectures). o April 19: Solutions to HW9, HW10, HW11 posted. o April 19: Some practice examples with solutons posted (about Galois theory). Weekly homework and other online material Homework 1: (due Wednesday Jan. 15) Section 18: 12, 24, 38, 55 Section 19: 10, 14, 26, 29 Homework 1 Solutions Homework 2: (due Wednesday Jan. 22) Section 20: 2, 29 Section 21: 13 Section 22: 31(c) Section 23: 10 Homework 2 Solutions Homework 3: (due Friday Jan. 31) Section 23: 14, 16, 34 Hint for 16: If a cubic polynomial is reducible it must have a zero. Use Theorem 23.11 and Corollary 23.12. Hint for 34: Consider the cases p=2 and p an odd prime separately. Show x^p+a has a zero in each case. Section 24: 6 Section 26: 2, 20 Hint for 20: Briefly discussed in class, use binomial theorem and show binomial coefficients (p choose i) are divisible by p. Homework 3 Solutions Homework 4: (due Monday February 10) Problem 1: Factorize the polynomial x^5+x^4+x^3+x^2+x+1 into irreducible factors over Q. Factorize this polynomial into irreducible factors over C. Problem 2: Let F be a field and E another field containing F. Let f(x), g(x) be polynomials in F[x]. Show that if f(x) is divisible by g(x) in E[x], i.e. there exists h(x) in E[x] such that f(x)=g(x)h(x), then f(x) is divisible by g(x) in F[x], i.e. h(x) is in F[x]. Section 26: 34, 37 Section 27: 8, 32, 35 Homework 4 Solutions Homework 5: (due Friday February 21) Section 29: 4, 18, 25, 36, 37. Homework 5 Solutions Homework 6: (due Friday February 28) Section 31: 4, 22 (here R(a+bi) means the smallest subfield of C containing R and a+bi, as in F(alpha)), 24, 28, 35 (Hint: note that 1 is not equal to -1 (why?), but for any b in F we have b^2= (-b)^2, conclude that there is some a in F which is not square of any b in F). Homework 6 Solutions Homework 7: (due Friday March 7) Section 31: 36 (hint: one way is to show that for any n>0 there are irreducible polynomials over Q with degree bigger than n), 37. Section 32: 1, 3. Homework 7 Solutions Homework 8: (due Friday March 21) Section 33: 6 (hint: recall material about cycle groups, see Section 6, Theorem 6.14 and Corollary 6.16), 9 (hint: show both fileds have the same number of elements), 11. Section 48: 4, 12. Homework 8 Solutions Homework 9: (due Wednesday April 2) Section 48: 32, 34, 36. Section 49: 6, 10 (let E be a subfield of algebraic closure of F). Section 50: 4. Homework 9 Solutions Homework 10: (due Wednesday April 9) Section 50: 8, 10, 22, 24 (moreover show that Q(z) is a Galois extension of Q, here z is a p-th root of unity in complex numbers and Q is the field of rational numbers, see Sec. 48, Exercie 36). Section 51: 4, 14. Homework 10 Solutions Homework 11: (due Wednesday April 16) Let E be the splitting field inside C of the polynomial f(x) = x^5 - 2, and let G=G(E/Q). (a) Show that E = Q(a, e) where a = 2^(1/5) and e = exp(2*pi*i/5). (b) Show that [E:Q] = 20, and deduce that |G| = 20. (c) Show that G is generated by the automorphisms theta and sigma given by: theta(a) = ea, theta(e) = e sigma(a) = a, sigma(e) = e^2. (Hint: determine all possible images of a and e under an automorphism in G, deduce that each of those possibilities gives rise to an actual automorphism, and that each such automorphism is of the form (theta^i sigma^j) with 0<= i <=4 and 0 <= j <= 3. (d) Verify that theta generates a cyclic subgroup of order 5, and that sigma generates a cyclic subgroup of order 4 in G. (e) Find the fixed field of the subgroup H generated by theta. Verify that H is a normal subgroup. HW11 solutions: see solutions to Practice Problems 2 (first problem) below. Some practice problems/examples about Galois theory: Practice problems 1 Solutions Practice problems 2 Solutions Midterm info sheet Solutions to midterm Final exam info sheet Suggested practice problems: Section 18: 6, 18, 19, 25, 27, 33, 35, 46, 52, 56 Section 19: 1, 2, 12, 17 Section 20: 3, 10, 23, 27 Section 21: 2, 4, 12 Section 22: 16, 20, 23, 24, 25, 29 Section 23: 1, 9, 17, 21, 25, 26, 36, Section 24: 4, 5, 9 Section 26: 1, 3, 10, 18, 22, 30, 32, 34, 37, 38 Section 27: 1, 5, 7, 10, 12, 14, 16, 18, 24, 28, 30, 32, 34, 35. Section 29: 1, 2, 5, 6, 7, 11, 12, 18, 23, 24, 26, 29, 30, 31, 32, 34, 36, 37. Section 31: 1, 3, 11, 19, 23, 24, 27, 31, 32, 33, 35, 36, 37, 38. Section 32: 2, 4, 7. Section 33: 5, 7, 8, 10, 11, 12, 13, 14. Section 48: 3, 5, 7, 9, 11, 13, 15, 15, 17, 19, 22, 29, 32, 34, 37, 38. Section 49: 3, 5, 6, 7, 8, 10, 11, 13. Section 50: 1, 3, 5, 7, 9, 14, 17, 18, 19, 21, 22, 23, 24. Section 51: 1, 4, 8, 9, 14, 15-21. Section 53: 1, 3, 5, 7, 8, 9, 11, 12, 15, 16, 17, 18, 19, 20, 22, 23. Section 54: 1, 4, 7, 8, 9, 10, 11, 12. Course Information Text: "A First Course in Abstract Algebra" by John Fraleigh, 7th Ed. (You should be OK using earlier editions, just make sure you compare the section numbers and exercise numbers with the 7th edition, few sections in eariler editions might be different from the 7th edition). Other references: "Abstract Algebra, An Introduction" by Thomas Hungerford "Galois Theory" by David Cox "Contemporary Abstract Algebra" by Joseph Gallian Course Outline: - January: Rings, fields, ideals and homomorphisms Sections: 18, 19, 21, 22, 23, 24 (only a brief discussion of quaternions), 26, 27 - February: Extension fields Sections: 29, 30, 31, 32, 33 - March and April: Automorphisms and Galois theory Sections: 48, 49, 50, 51, 53, 54, (sketch of 55, 56) Lectures: MWF 12:00PM - 12:50PM Location: G37 Benedum Hall Dates: January 6 to April 18, 2014 There will be weekly homeworks, a midterm and a final exam. Important dates: First class: January 6 Last class: April 18 Holidays: January 20, March 9-16 (Spring recess) Final exam period: April 21-26 Midterm: February 12 (Wednesday) Grading scheme: 25% homework + 25% midterm + 50% final exam. Lowest homework will be dropped. Instructor's Information Email: kavehk AT pitt.edu Office: Thackeray #424 Office hours: 11:00AM-12:00PM Tuesdays or by appointment. You can drop by I am usually in. Office phone: 412-624-8331 TA MAC hours: Thursdays 10-11, Fridays 11-12. |
