Modern Algebra II, Math 4613 and 5003, Section 001, Fall 2007
Syllabus
Office hours: Mon,Wedn 1:30-3;Tu, Th
12-1:30.
Final Exam on Tuesday,
April 29, 10 am in class.
Hw 9 (Due Friday April
25): 13.4: 1,3; 13.5: 2,5.
Hints: 13.4, #1, very
similar to example in class, find all complex roots of x^4-2 and show the
splitting field is generated by
2 elements over Q, then
apply techniques from class to count the degree.
#3, note
x^6-1=(x^2-1)(x^4+x^2+1), this should help to find the complex roots, and for
degree note x^4+x^2+1 is not irreducible.
13.5, #2, to find
irreducible polynomials check for roots in F_2 and for degree 4 use the fact
that
there is only one irreducible
polynomial of degree 2; #5, using 1st approach note that
[F_p(alpha):F_p] is the degree of irreducible polynomial dividing x^p-x+a, show
that every irreducible factor has the same degree and find contradiction.
Hw 8 (Due Friday April
18): 13.1: #1,8; 13.2: #4,5,14; 13.3: #4.
Hints: #1 is similar to
Example (4) in the book; 13.2: #4, 1st one is easy, for the second
show Q(1+2^(1/3)+4^(1/3)) is included
in Q(2^(1/3)) and 1+2^(1/3)+4^(1/3)
is not in Q (why? use basis), then apply tower law.
#5 show that x^3-2 does
not have a root in Q(i), note i=sqrt(-1) is a root of x^2+1, so we know degree
of Q(i) over Q,
then use tower law.
#14, Note F(alpha^2)
subfield of F(alpha). Use the fact that alpha is a root of polynomial
x^2-alpha^2 from F(\alpha^2)[x],
then apply tower law.
Exam 2 is on Friday,
April 11, 9:05 am. (covers sections on
modules 10.1-10.4, 12.1)
Homework 7 (Due April
7): 12.1, #1,2,5,7,8; Sec 10.4: #2,27.
Hints: 12.1, all hints
in the book; for #7, you can assume that direct sum is direct product, show
that isomorphism as R-modules; for #8, one possibility is to solve the problem
when B=Rb (generated by b), you will need to use the property of
PID: nonzero prime ideal
is maximal, then relate Ann(B) to Ann(Rb).
10.4, #2, to show
nonzero construct nonzero homomorphism from the second Z-module to Z/2Z.
27(d), C-algebra has an
induced structure of C-module (check definition), you need to show Phi is
C-linear (C-module homomorphism).
Homework 6 (Due Wed
March 12): 10.3: #4,5,7,11,12,13.
Hints: #4, to give
example look for a quotient of rational numbers Q or anything similar;
#5, for each generator
of m_i of M there is such r_i, construct r from these; example from #4 will
work here;
7, need to show that
every element in M is a linear combination of a finite # of elements; construct
these elements explicitly form the generators of N and M/N.
11, follow the book
hint; for 2nd part, note that isomorphism has inverse homomorphism.
12, explicitly construct
natural homomorphisms; for instance in part (a), send map \phi: A\times B --
>M to the pair
(\phi_1, phi_2) such
that \phi_1(a)=\phi(a,0), \phi_2(b)=\phi(0,b). Check that you get R-module
isomorphism.
Similarly try part (b).
13, assume F=R^n then
apply 12(a) and #10 from previous section or try to construct explicitly the
map like in 12(a).
Homework 5 (Due Wed.
March 5): 10.1: 8,9,10,11; 10.2: 6,8,10.
Hints: 8(b), there are
plenty of examples, try R=Z/6Z; 9, do not assume that R is commutative; 11(b)
every finite abelian group is a direct product of cyclic groups; first, compute
annihilator explicitly as an abelian subgroup of M;
10.1,#6 as in class
construct homomorphism Z -- >Hom(Z/nZ,Z/mZ), show: well-defined, surjective,
ker=...
10, R commutative with
1, consider map R-- > Hom (R,R), r maps to linear map which is
multiplication by r.
Homework 4 (Due Mon Feb
18): Sec 9.3, #1,3; Sec 9.4, #9,11,13,14.
Hints: Hints: 9.3, #1, Read the end of Section 9.3 (a monic
polynomial
leading coefficient = 1). Restate equivalently: Suppose R is
UFD,
p(x)=a(x)b(x) in R[x] with a(x),b(x) in F[x] monic, then
show that
a(x) and b(x) in R[x]. (As in class a(x)=r*a'(x),
b(x)=s*b'(x)
where a'(x),b'(x) in R[x] with GCD of coeffic. =1, and r,s
in F
with r*s in R. Why r,s must be in R? Think about the case
R=Z
integers, F=Q rationals. With
Z[2sqrt(2)]={a+b2sqrt(2)|a,b\in Z}
show that its field of fractions is
Q[2sqrt(2)]={a+bsqrt(2)|a,b\in
Q} (just consider inverses in real numbers and check that
sqrt(2)
in Q[2sqrt(2)]), then factor x^2-2 in Q[2sqrt(2)][x].
#3, Why x^2 and x^3 are irreducible in R?
9.4, #9 First show that sqrt(2) is irreducible in Z[sqrt(2)]
using
norm N(a+b*sqrt(2))=a^2-2b^2 which is multiplicative (we did
similar calculations in Z[\sqrt(-5)]. To show irreducibility
of
x^2-\sqrt(2) either use the fact Z[sqrt(2)] is UFD, whence
irreducible is prime and apply Eisenstein's Criterion
Another way:
assume reducibility x^2-\sqrt(2)=(x-A)(x-B) with A,B in
Z[sqrt(2)]
(Explain why you can do this way), and use the norm to get a
contradiction.
#11, Apply Eisenstein to Q[y][x]; #13, apply rational roots
criterion (proposition 11); #14 Techniques of factorization
of low
degree polynomials covered in class, you must show that the
factors are irreducible by using all available criteria.
Homework 3 (Due Wedn,
Feb 6): Sec 8.3, #6(a,b),8(a,b), Sec 9.1,#7,13,17,
Sec 9.2, #3,7.
Hints: 8.3,#6(a), reduce cosets using 1+i=0 and i^2=-1 as in
class; 6(b), need to show that Z[i]/(q) has exactly q^2
elements
(similar to part (a)), then show that q is prime in Z[i] by
propositions in class, deduce that (q) is maximal. 8(a), we
showed in class: 1+sqrt(-5) is irreducible (you can skip
it),
similarly do the rest, second part is just an obvious
statement
that factorization in R is not unique (not UFD); 8(b), we
showed
I_2 is prime (skip it), similarly do I_3 and I'_3 (or use
hint in
the book); 9.1, 7: try example in #6; 13, one is an integral
domain and the other one is not, show that map F[x,y]-- >
F[y], x-->y^2, y-->y,
induces isomorphism of F[x,y]/(x-y^2) onto F[y],
to show injectivity prove that every coset is uniquely
represented
by a polynomial dependent on y only. 17, we'll do some part
in
class; 9.2,#7 reducing mod 2 and using x^3=-1 show that
there are
precisely 8 cosets, whence there is only a finite number of
possible ideals, find all the distinct ideals (how many?),
you
should be able to work yourself with such a small ring.
Homework 2 (Due Mon Jan
28): Sec 7.5, #2,3; Sec 7.6, #1,3; Sec 8.1, #7,10; Sec 8.2, #2,8.
Hints: 7.5, #2 quotient field is field of fractions S^{-1}R
with
S=R\{0}, construct natural homomorphism from D^{-1}R to
S^{-1}R, show it's well-defined, injective;
#3 you can use Exercise 26 and 28 (sec7.3) giving 2 cases
F is of prime characteristic p or of characteristic 0.
In 1st case p*1:= 1+...+1=0 (p times)
in field F, and F_0 should contain p elements (explicitly
describe
them). Similarly, if F has characteristic 0, then m=1+...+1
(m
times) is never 0. Desribe elements of F_0 explicitly.
For uniqueness note that a subfield of F should always
contain 0 and 1
from F. 7.6, #1, R need not to be commutative. Show that the
map
Re x R(1-e) --> R, (x_1e,x_2(1-e))--> x_1e+x_2(1-e) is
a bijective
homorphism. 8.1,#10, use the hint in the book and then show
that
infinite sequence with N(a_1)=N(a_2)=N(a_3)=...and
representatives
a _i of distinct cosets of I is not possible (recall
N(a)=|a|^2).
Homework 1 (Due Jan.16): Sec 7.3, #24,34,35; Sec 7.4, #10,11,37
Hints: 7.3, 24 straightforward, try example in (b) with
inclusion homomorphism;
34(c), try integers ring Z; (d) assume R has 1, and write
1=a+b, where a in I, b in J;
35, easy; 7.4, note 0 in P, and do by definitions; 11 easy;
37, assume R has 1, let x be in R\M, consider ideal Rx, note Rx+M=??? and use
proposition from class that every proper ideal is contained in a maximal ideal
to show that Rx=R, 2nd part is easier.