Abstract Alg Final

Question 1

prove H is a normal subgroup of G (with some A in G, B in H)

Answer 1

show ABA^-1 is in H, which implies AHA^-1 is a subgroup of A

Question 2

fundamental homomorphism theorem

Answer 2

let G->H be a homomorphism. then G/ker(f) ~= f(G)

Question 3

prove isomorphic

Answer 3

f(uv)=f(u)f(v), its surjective

Question 4

prove A subring of B

Answer 4

A subset of B, show A nonempty, x-y in A and xy in A

Question 5

little theorem of fermat to show remained when 26^3250 divided by 17

Answer 5

1. 17 is prime, 17 does not 26 so [26^17-1]=[1]
2. split 3250 up so you have x*(17-1)+y
3. then [26^x16][26^2] = [1]*[simplified 26]^2
4. = [simplified 26]^2
5. factor out the 17 term so youre left with one term which is the remainder

Question 6

when things aren’t isomorphic

Answer 6

1. if one ring has unity and the other doesnt
2. if one isnt a field and the other is
3. if one has zero divisors and the other doesnt

Question 7

show I is an ideal of S

Answer 7

if P,Q, in I and A in S. Show P-Q in I, AP in I, PA in I, and 0 is in I

Question 8

show f is a ring homomorph from S to R

Answer 8

pick A in S.

1. f(A) is in R so well defined
2. show f(A+B)=f(A)+f(B)
3. show f(AB)=f(A)f(B)

Question 9

ker(f)=

Answer 9

{A in S: f(A) = 0 or identity element}

Question 10

commutative ring

Answer 10

if a,b in R, then ab=ba

Question 11

unity

Answer 11

if there is an identity element

Question 12

division ring

Answer 12

has unity, if identity element does not equal zero, and every element is a unit

Question 13

integral domain

Answer 13

ab=ba, r has unity, unity does not equal zero, has no zero divisors

Question 14

field

Answer 14

ab=ba, r has unity, unity does not equal zero, every element is a unit, has no zero divisors

Question 15

unit of R

Answer 15

if u is an element in R, then u has an inverse in R

Question 16

zero divisor

Answer 16

r in R. if there exists a w in R such that rw or wr = 0 , r is a zero divisor

Question 17

Zr X Zs ~= Zrs iff

Answer 17

gcd(r,s)=1

Question 18

A is a ring if

Answer 18

1. closed
2. associative and distributive
3. ab=ba for the group A

Question 19

u is not a zero divisor when

Answer 19

ring with unity and u is a unit

Question 20

characteristic of R

Answer 20

for a in R, if na=0 then n is a characteristic of r

Question 21

ring isomorphism from (R,+,*) to (R,&,%)

Answer 21

f(a+b)=f(a)&f(b) and f(a*b)=f(a)%f(b) and bijection

Question 22

f is irreducible if

Answer 22

f does not have a factor g such that 1

Question 23

if Z a subring of R and R has no zero divisors,

Answer 23

Z has no zero divisors

Question 24

things that are fields

Answer 24

R, C, Q

Question 25

things that are int domains

Answer 25

Z, R, Q, C