MTH 819
Algebra II
S13
Final Exam/Solutions
# 1. Let K F be a eld extension. Suppose char F = 3 and F = K(t), where t F is transcendental over K.
Put E = K(t3 ). Let V be a 2-dimensional vector space over F and A EndF (V ). Suppose that the Jordan
Canonic
MTH 819
Algebra II
S13
Practice Final Exam . Solutions
# 1. Let F be an algebraic closure of Z3 and for n Z+ let F3n be the unique subeld of order 3n in F.
(a) Determine all subelds of F312 .
(b) For each subeld K of F312 determine AutK (F312 ).
Proof. (a
MTH 819
Algebra I
S13
Homework 7/ Solutions
Denition A. A ring R is called semisimple if there exists a family (Ri )iI of ideals in R such that R =
and each Ri , i I is a simple ring.
iI
Ri
# 1. Let R be a Artinian ring. Show that R is semisimple as ring
MTH 819
Algebra I
S13
Homework 6/ Solutions
For the rst two homework problems let R be a ring, X a right R-module, Y a left R-module, D an abelian
group, s X Y D an R-balanced function, A X , B Y and S R. Write xy for s( x, y) and dene
A = cfw_y Y ay = 0
MTH 819
Algebra I
S13
Homework 5/ Solutions
Denition A. Let K F be a nite eld extension and r a prime. K F is called a r-extension if dimK F = rn
for some n N. K F is called an r extension if r does not divide dimK F.
# 1. Let K F be a nite Galois extensi
MTH 819
Algebra I
S13
Homework 4/ Solutions
# 1. Let K F be a nite eld extension with p = char K 0. Suppose that p does not divide dimK F. Show that
K F is separable.
Proof. Let b F and f the minimal polynomial of f over K. By Lemma 4.2.23(f),(g) there ex
MTH 819
Algebra I
S13
Homework 3/ Solutions
# 1. Let S R be a ring homomorphism and so R is an S -ring (see Remark C on Homework 2). Let V
and W be a free S -module with basis v = (vi )iI and w = (w j ) j J respectively. Let f HomS (V, W ) and let
A = Mvw
MTH 819
Algebra I
S13
Homework 2/ Solutions
# 1. Let R be an integral domain and M a divisible R-module. Show that T( M ) is a divisible R-module.
Proof. Let m T( M ) and 0 r R. Since M is divisible there exists v M with rv = m T( M ). Then
r(v + T ( M )
MTH 819
Algebra I
S13
Homework 1/ Solutions
Denition A. Let R be PID and V a unitary R-module. Let p be a prime in R and n Z+ . Then
d p,n (V ) = dimR
Rp
pn1 AnnV ( pn ) pn AnnV ( pn+1 )
Note here that pn1 AnnV ( pn ) pn AnnV ( pn+1 ) is a module over the
MTH 819
Algebra II
S13
Final Exam
justify all your answers
Your Name:
#1. Let K F be a eld extension. Suppose char F = 3 and F = K(t), where t F is transcendental over K.
Put E = K(t3 ). Let V be a 2-dimensional vector space over F and A EndF (V ). Suppos
MTH 819
Algebra II
S13
Practice Final Exam
justify all your answers
#1. Let F be an algebraic closure of Z3 and for n Z+ let F3n be the unique subeld of order 3n in F.
(a) Determine all subelds of F312 .
(b) For each subeld K of F312 determine AutK (F312