HOMEWORK #8
SOLUTIONS TO SELECTED PROBLEMS
Problem 8.4 Normality of the composite. Let K L1 , L2 K be
two nite extensions.
Lemma 1. If L1 /K and L2 /K are normal, then so is their composite
L1 L2 /K.
First proof. We know that a nite extension L/K is norma
HOMEWORK #9
SOLUTIONS TO SELECTED PROBLEMS
Problem 9.5. Let K be of prime characteristic p, and let L/K be Galois
extension of degree p. Since [L : K] = p, the Galois group of L/K is cyclic
of order p. Let be a generator.
First reduction. Suppose that the
SOLUTIONS TO QUIZ 1
Question 1. If f (x) F [x] is a polynomial over a eld F and 0 = a, b F
are scalars, then f (x) is irreducible if and only if f (ax + b) is irreducible.
So to prove irreducibility it is enough to consider an appropriate linear
substitut
SOLUTIONS TO QUIZ 2
Question 1. Let g K[t] be the minimal polynomial of over K. Then
we know that [K() : K] = deg g. On the other hand, is a root of f ,
hence by minimality of g we have that g divides f .
If f = g we see immediately that [K() : K] = deg f
HOMEWORK #10
SOLUTIONS TO SELECTED PROBLEMS
Problem 10.6. The idea is to use the theorems that show the existence of
elements with special cycle structure (when viewed as permutation on the
roots). Here are a few examples.
The polynomial f (t) = t5 4t + 2
HOMEWORK #7
SOLUTIONS TO SELECTED PROBLEMS
Problem 7.1 Separability of towers. We prove the following:
Proposition 1. Let L/K be a nite extension and let K F L be an
intermediate eld. Then L/K is separable if and only if L/F and F/K are
separable.
Proof.
HOMEWORK #4
SOLUTIONS TO SELECTED PROBLEMS
Problem 4.2. The derivation D : K[t] K[t] is dened by D(tn ) = ntn1
and then extending by linearity. To prove (a), it is enough to consider the
basis elements tn of K[t] over K. Indeed, one has
D(tn tm ) = (n + m
HOMEWORK #3
SOLUTIONS TO SELECTED PROBLEMS
Problem 3.3. (a) Use Eisensteins criterion with the prime 7. 7 does not
divide the highest coecient, does divide all other coecients and 72 does
not divide the constant term.
(b) Use the following lemma:
Lemma. L
HOMEWORK #5
SOLUTIONS TO SELECTED PROBLEMS
Problem 5.3 Construction of nite elds. Let p be a prime and let
Fp be the eld with p elements. Let r 1 be an integer and set q = pr .
We will construct a eld with q elements and prove its uniqueness (up to an
iso
HOMEWORK #2
SOLUTIONS TO SELECTED PROBLEMS
p i pi
. All the binomial coefProblem 2.1. One has (a + b)p = p
i=0 i a b
p
cients i are divisible by p for 0 < i < p, as they have p factor in the
nominator and no p factor in the denominator.
Problem 2.2. The p
HOMEWORK #1
SOLUTIONS TO SELECTED PROBLEMS
Problem 1.2. (a) To compute [C : R], note that C = R(i) and i is a solution
to the polynomial x2 + 1 R[x]. This polynomial is irreducible, otherwise
it would have a linear factor hence a solution in R, which is i