MATH30010: Field Theory
Homework 5: Solutions
1. Let L1 /F and L2 /F be eld extensions. Suppose that : L1 L2
is a eld embedding satisfying (a) = a for all a F (we say that
xes F ). Suppose that b L1 is a root of q(x) F [x]. Prove that
(b) L2 is also a ro
MATH30010: Field Theory
Mid-term Test (2010)
Solutions/Comments
1. Dene what is meant by a eld.
Solution: From class
2. Show that in any eld F , 0 a = 0 for all a F .
Solution: From homework 1.
3. What is the order of 3 in the eld F11 ?
Solution. In F11 ,
MATH30010: Field Theory
Homework 3: Solutions
1. Let K be a eld and let F be a subeld (i.e. K/F is an extension of
elds). Let p(x), q(x) F [x] with q(x) = 0. Prove that if q(x) divides
p(x) over K, then q(x) divides p(x) over F .
[Hint: Use the division a
MATH30010: Field Theory
Homework 1: Solutions
1. Prove the uniqueness of multiplicative inverses in any eld F ; i.e., let
a F and let b, c F satisfy
a b = a c = 1.
Using the eld axioms, show that b = c.
Solution: We have c = 1 c = (b a) c = b (a c) = b 1
MATH30010: Field Theory
Homework 4: Solutions
1. Let
n := e2i/n = cos
2
n
+ i sin
2
n
C.
(So (n )n = 1.) Explain why n is an element of (multiplicative) order
n in C.
Show that the minimal polynomial over Q of p , for p prime, is 1 +
x + + xp1 .
n
Soluti
MATH30010: Field Theory
Homework 2: Solutions
1. Hamiltons quaternions, H, consists of the set of all symbols of the
form a + bi + cj + dk, where a, b, c, d R and i, j, k are symbols
satisfying the identities
i2 = j 2 = k 2 = 1,
ij = k = ji.
Addition is d