Herstein, Chapter 3.

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
*xy ^{2}-y-1, x -y^{} *.

2. Does the ideal from the previous exercise contain a nonzero polynomial

3. Solve the system of equations

xy-x-y=2

xy^{2}-2x-y=2

in complex numbers (find all solutions).

** HW 5. Due Sunday, February 18, 2018 **

For every set *V* of points in *ℂ ^{n}* let

1. Show that

2. Let

3. Let

4. Let

5. Using a Gröbner basis solve the following system of polynomial equations:

-x^{2}y^{2}-x^{2}+xy-y^{2}=0

-x^{2}z^{2}-x^{2}+xz-z^{2}=0

-z^{2}y^{2}-z^{2}+zy-y^{2}=0

** 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/b ^{d}*.

From now till Problem 5 assume conditions of problem 2.

3. Suppose that

4. Deduce from 2 and 3 that

5. Deduce from 4 that the decimal fraction

6. Let

1. Prove that every polynomial with real coefficients and odd degree has at least one real root.

2. Prove that the polynomial

3. Let

4. Exercise 9 from Herstein's book, page 236.

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.

Below we always assume that the field

1. Let

(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 *r _{i}, i=1,...,n*. Let

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

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

6. Find the Galois group of the polynomial

1. Problem 7, page 259 (Herstein's book)

2. Describe all subfields of the field

3. Show that the group of automorphisms of the field of real numbers fixing all rational numbers is trivial.