1. p 130, # 6
2. p 135, # 3
3. p. 135, #5 and #6
4. p. 136, #12.
5. p. 143, # 6.
HW 2. Due Sunday, January 28, 2018
1. p. 149, # 2
2. p. 152, # 2
3. p. 152, # 3 (a),(b)
4. p. 152, # 8
5. p. 158, #2 (a), (c)
6. p. 158, #6.
HW3. Due Sunday, February 4, 2018.
1. p. 166 #6 .
2. p.167, #10.
3. p. 167, #14.
4. p. 167. #16.
5. p. 167, #18.
6. p. 167, #28
HW 4. Due Sunday, February 11, 2018
There are many texts on the Web about Gröbner bases and the Hilbert basis theorem. For example, https://www.kent.ac.uk/smsas/personal/gdb/MA574/week4.pdf .
1. Find the Gröbner basis of the ideal of K[x,y] (K is a field of characteristic 0) generated by polynomials xy2-y-1, x
2. Does the ideal from the previous exercise contain a nonzero polynomial f(y)?
3. Solve the system of equations
in complex numbers (find all solutions).
HW 5. Due Sunday, February 18, 2018
For every set V of points in ℂn let I(V) be the ideal of all polynomials from ℂ[x1,...,xn] that vanish on V. Conversely for every ideal I of ℂ[x,...,xn] let V(I) be the algebraic variety of all roots of this ideal.
1. Show that I(V(I)) is the radical of I (that is the ideal of all polynomials whose large enough powers are in I) for every ideal I.
2. Let f,g be two polynomials in n variables and complex coefficients. Suppose that f is irreducible and V((f)) is contained in V((g)). Prove that f divides g.
3. Let K be any field and consider the ring of polynomials in variables x1,...,xm,y1,...,yn over K. Let G be a Gröbner basis of some ideal I with respect to the order x1 > ... >xm > y1 >... > yn and Lex. Let S be the ring K[y1,...,yn]. Show that G ∩ S is a Gröbner basis of the ideal I ∩ S of S.
4. Let V be the set of all points in ℂ[x,y,z] of the form (t3,t 4, t5) where t is an arbitrary complex number. Find a Gröbner basis of the ideal I(V). Hint: use Exercise 3.
5. Using a Gröbner basis solve the following system of polynomial equations:
HW 6. Due Sunday, February 25
1. Let G be a group, and a ≠ 1 an element of G of finite order. Show that for every ring R, the element 1-a is a zero divisor in the group ring R[G].
2. Let C be the cyclic group of order 3 and F be the field with 3 elements. Find all units (invertible elements) and zero divisors in the group ring F[C].
3. Let R be a Euclidean domain and M be a finitely generated module which is not a direct sum of proper submodules. Show that either M is isomorphic to R or for some prime element p of R and some power q of p the module M is isomorphic to the R-module R/(q).
HW 7. Due Sunday, March 11
The best proofs of the fundamental theorem of algebra and the fact that e is transcendental can be found in the highly recommended book by Jim Cannon Two-dimesional spaces, V. 1-3
1. Prove that if a is a transcendental complex number (over ℚ), then the field ℚ(a) is isomorphic to the field of rational fractions in one variable ℚ(x).
2. Let α be an algebraic irrational number and p(x) be the minimal polynomial with integer coefficients having α as a root. Let d be the degree of p(x). Show that for every reduced rational fraction a/b we have |p(a/b)| ≥ 1/bd.
From now till Problem 5 assume conditions of problem 2.
3. Suppose that |a/b - α| < 1 . Prove that for some constant c depending only on α we have p(a/b) < |a/b -α|/c. Hint: Use the polynomial q(x) such that p(x)=q(x-α).
4. Deduce from 2 and 3 that |α - a/b| >c/(b d) for every reduced fraction a/b.
5. Deduce from 4 that the decimal fraction α=0.1010010000001..., where the number of 0's between digit 1 number n and digit 1 number n+1 is n!, is transcendental.
6. Let K be a subfield of the field ℂ. Show that the set Fof complex numbers that are algebraic over the field K is a subfield of ℂ and every root of a polynomial with coefficients in F belongs to F.
HW 8. Due Sunday, March 18
1. Prove that every polynomial with real coefficients and odd degree has at least one real root.
2. Prove that the polynomial x 5-3x-1 has exactly 3 real roots. (Do not use graphs.)
3. Let a be a root of polynomial x 2 -x+1 and b be a root of the polynomial x2 +x-1. Find a polynomial of the smallest degree with integer coefficients having root a+b.
4. Exercise 9 from Herstein's book, page 236.
HW 9. Due Sunday, March 25
1. Problem 7 on page 249 of Herstein's book.
2. Problem 8 (c) on page 249.
3. Problem 9 (c) on page 249.
4. Problem 12 on page 249.
5. Problem 18 on page 250.
HW 10. Due Sunday, April 8
Below we always assume that the field K is of characteristic 0.
1. Let p(x) be a polynomial over the field K without multiple roots.
(a) Show that the Galois group of the splitting field of p(x) acts faithfully (i.e., with trivial kernel) on the set S of roots of p.
(b)Show that the action is transitive if and only if p(x) is irreducible.
2. Suppose the polynomial p as above has the set of roots ri, i=1,...,n. Let D be the square of the product of all differences ri - rj, i > j . Show that D belongs to K. D is called the discriminant of p.
3. Find the discriminant of a quadratic polyniomial (in terms of the coefficients).
4. Show that if a square root of the discriminant of the polynomial p(x) of degree n is in K then the Galois group of that polynomial is a subgroup of the alternating group An. , that is, it consists of even permutations of the roots.
5. Show that the Galois group of a cubic irreducible polynomial is either cyclic of order 3, if the discriminant of the polynomial has a square root in K, or isomorphic to the symmetric group on 3 symbols otherwise.
6. Find the Galois group of the polynomial x3 - 6x-3 over the field of rational numbers.
HW 11 . Due Sunday, 4/22/2018
1. Problem 7, page 259 (Herstein's book)
2. Describe all subfields of the field ℚ[ √ 2 , √ 3 ] as ℚ[t] for some numbers t. Which of these fields are normal?
3. Show that the group of automorphisms of the field of real numbers fixing all rational numbers is trivial. Hint: Use the fact that positive real numbers have real square roots to show that every automorphism of real numbers must take positive numbers to positive numbers. Then show that the automorphism must preserve the order of real numbers. Finally trap every real number between close pairs of rational numbers.