UNIVERSITY OF DUBLIN
XMA2215
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
Trinity Term 2011
E
SF Mathematics
SF Two Subject Moderatorship
PL
MA2215 Fields, rings and modules
2 hours
Dr. R. Levene
Credit will be giv
Mathematics 2215: Rings, elds and modules
Solutions to tutorial exercise sheet 1
Let R and S be rings.
1. If R S, show that R is unital if and only if S is unital.
Solution Since R S, there is an isomorphism : R S.
Suppose R is unital. We claim that (1R )
Some useful facts about vector spaces over elds
Let F be a eld.
Denition. A vector space over F is a set V with two maps, + : V V V and : F V V
so that:
(a) (V, +) is an abelian group
(b) for all , F and all v, w V we have
(v + w) = v + w,
( + ) v = v +
A reminder of some useful group theory
Let (G, +) be an abelian group and let H be a subgroup of G.
1. If x G then the right coset of H by x is the set
H + x = cfw_h + x : h H .
2. When are two of these cosets equal? Answer:
H + x = H + y x y H .
3. We wr
7. (a) We have C , so x is the minimum polynomial of over C.
(b) Since C we have R() C. Since R we have [R() : R] > 1. Now [C : R] = 2, so
2 = [C : R] = [C : R()] [R() : R] > [C : R()], so [C : R()] < 2, so [C : R()] = 1, so
R() = C.
1
1
1
1
We have = cos
MA2215 20102011
A (non-examinable) proof of Gauss lemma
We want to prove:
Gauss lemma. If R is a UFD, then R[x] is a UFD.
We know that if F is a eld, then F [x] is a UFD (by Proposition 47, Theorem 48
and Corollary 46). In outline, our proof of Gauss lemm
Mathematics 2215: Rings, elds and modules
Solutions to homework exercise sheet 1
1. Show that Z[i] = cfw_a + bi : a, b Z is a unital commutative ring under addition and multiplication of complex numbers. [This is called the ring of Gaussian integers]. Doe