Algebra general exam. January 8, 2020, 9am -1pm

Your UVa ID Number:

Directions.

Please show all your work and justify any statements that you make.
State clearly and fully any theorem you use.
Vague statements and hand-waving arguments will not be appreciated.
You may assume the statement in an earlier part proven in order to do a later part.
DO EACH PROBLEM ON A SEPARATE SHEET OF PAPER, AND STAPLE THEM TOGETHER IN THE CORRECT ORDER BEFORE TURNING THE EXAM IN.

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Problem 1

(a) ( 7 pts) Let

C

be a cyclic group of order

n \geq 2

. Explain briefly why

\operatorname{Aut}(C) \cong \mathbb{Z}_{n}^{\times}

(the multiplicative group of

\mathbb{Z}_{n}

) and why

\operatorname{Aut}(C) \cong \mathbb{Z}_{n-1}

n

is prime.
(b) ( 7 pts) Let

G

be a finite group, let

p

be the smallest prime dividing

|G|

, and suppose that

G

contains a normal subgroup

C

of order

p

. Prove that

C

lies in the center of

G

Problem 2

(10 pts) Let

A

be the subgroup of

\mathbb{Z}^{2}

generated by

(2,6)

and

(4,8)

. Prove that

A \cong \mathbb{Z}^{2}

and find (with proof) elements

v_{1}, v_{2} \in \mathbb{Z}^{2}

and

n_{1}, n_{2} \in \mathbb{N}

such that

\mathbb{Z}^{2}=\mathbb{Z} v_{1} \oplus \mathbb{Z} v_{2}

and

A=\mathbb{Z}\left(n_{1} v_{1}\right) \oplus \mathbb{Z}\left(n_{2} v_{2}\right)

Problem 3

(10 pts) Let

R=\mathbb{Z}[i]

, the ring of Gaussian integers. Find (with complete proof!) the number of ideals of

R

which contain 30 .

Problem 4

Let

R

be a commutative ring with 1 and let

x \in R

.
(a) (3 pts) Prove that

x \notin R^{\times}

if and only if

x \in M

for some maximal ideal

M

R

.
(b) ( 9 pts) Prove that the following are equivalent:
(i)

x \in M

for every maximal ideal

M

R

(ii)

1+x y \in R^{\times}

for every

y \in R

Problem 5

(12 pts) Let

F

be a field and let

A, B \in \operatorname{Mat}_{n}(F)

for some

n \in \mathbb{N}

. Suppose that

A^{2}=B^{2}=I

and

\operatorname{rk}(A-I)=\operatorname{rk}(B-I)

. Prove that

A

and

B

are similar. Hint: Consider separately the cases

\operatorname{char} F \neq 2

and

\operatorname{char} F=2

Problem 6

In each part of this problem determine if the given objects are isomorphic:
(a)

(4 \mathrm{pts}) \mathbb{R}

and

\mathbb{C}

\mathbb{Q}

-vector spaces
(b)

(3 \mathrm{pts}) \mathbb{C}

and

\mathbb{R} \times \mathbb{R}

as rings
(c) ( 4 pts)

\mathbb{R}[x] /\left(x^{2}-1\right)

and

\mathbb{R} \times \mathbb{R}

as rings
(d) ( 4 pts)

\mathbb{R}[x] /(x-1)

and

\mathbb{R}[x] /(x+1)

\mathbb{R}[x]

-modules

Problem 7

Let

F, K_{1}, K_{2}

and

L

be fields with

F \subseteq K_{1} \subseteq L

and

F \subseteq K_{2} \subseteq L

. Recall that the tensor product

K_{1} \otimes_{F} K_{2}

has a natural structure of a ring where multiplication of simple tensors is given by

(a \otimes b) \cdot(c \otimes d)=a c \otimes b d

for all

a, c \in K_{1}

and

b, d \in K_{2}

.
(a) ( 5 pts) Prove that there exists a unique ring homomorphism

\pi

K_{1} \otimes_{F} K_{2} \rightarrow L

such that

\pi(a \otimes b)=a b

for all

a \in K_{1}, b \in K_{2}

.
(b) ( 2 pts) Assume that

K_{1} \otimes_{F} K_{2}

is a field. Prove that

\pi

is injective.
(c) ( 5 pts) Assume that

K_{1} \cap K_{2} \neq F

. Prove that

\pi

is not injective.

Problem 8

Let

K / F

be a field extension. Suppose that

K=F(a, b)

for some

a, b \in K

such that

a^{2} \in F

and

b^{2} \in F

.
(a) ( 3 pts ) Prove that

[K: F] \leq 4

(b) ( 4 pts ) Give a specific example (with full proof) where

[K: F]=4

\operatorname{char}(F) \neq 2

. Prove that the extension

K / F

is Galois
(d) ( 4 pts ) Now assume that

F

is finite. Prove that

[K: F] \leq 2