Caltech
Math 5c
Spring 2013
Homework 6
Solutions
Problem 1 [14.2.23] Let K be a Galois extension of F with cyclic Galois group of order n
generated by . Suppose K has NK/F () = 1. Prove that = for some nonzero K .
Proof. By the linear independence of the
Caltech
Math 5c
Spring 2013
Homework 7
Solutions
Problem 1 [14.6.35] Prove that the discriminant D of the polynomial xn + px + q is given by
(1)n(n1)/2 nn q n1 + (1)(n1)(n2)/2 (n 1)n1 pn .
Proof. Let 1 , . . . , n be the roots of x) = xn + px + q . Recall
Caltech
Math 5c
Spring 2013
Homework 5
Solutions
Problem 1 [14.2.3] Determine the Galois group of (x2 2)(x2 3)(x2 5). Determine all the
subelds of the splitting eld of this polynomial.
Solution. It is easy to see that K = Q( 2, 3, 5) the splitting eld of
Caltech
Math 5c
Spring 2013
Homework 4
Solutions
Problem 1 [14.1.7]
(a) Prove that any Aut(R/Q) takes squares to squares and takes positive reals to positive reals.
Conclude that a < b implies (a) < (b) for every a, b R.
1
1
1
(b) Prove that m < a b < m i
Caltech
Math 5c
Spring 2013
Homework 3
Solutions
Problem 1 [13.2.18] Let k be a eld and let k (x) be the eld of rational functions in x with
P
coecients from k . Let t k (x) be the rational function Q(x) with relatively prime polynomials
(x)
P (x), Q(x) k
Caltech
Math 5c
Spring 2013
Homework 2
Solutions
Problem 1 [13.2.14] Prove that if [F () : F ] is odd then F () = F (2 ).
Proof. If F (2 ) then F (2 ) is a proper subeld of F (). Moreover satises x2 2 F (2 ),
/
so [F () : F (2 )] = 2. However,
[F () : F ]
Caltech
Math 5c
Spring 2013
Homework 1
Solutions
Problem 1 [13.1.5] Suppose is a rational root of a monic polynomial in Z[X ]. Prove that Z.
Proof. By the rational root theorem (Prop. 11, Ch.9) if =
and (p, q ) = 1 then q | 1 and therefore Z.
p
q
Q is a
Math 5c
Final
Due Thursday, June 14, 12 pm
Do the following 5 problems in 4 hours. You may cite any theorems proved in class or Dummit and
Foote, but prove any material you use from other sources.
You are allowed to use encyclopedic on line resources (E.g
Ma 5c, Spring 2012
Caltech
Problem Set 1
Solutions
[13.1.5] By the rational root theorem (Prop. 11, Ch.9) if =
monic polynomial and (p, q ) = 1 then q | 1 and therefore Z.
p
q
Q is a root of the
[13.1.7] If a cubic polynomial factors then one of its fact
Caltech
Ma 5c, Spring 2012
Problem Set 2
Solutions
[13.3.1] Note that a regular 9-gon has an exterior angle of 360 = 40 . Since the only
9
constructible angles are precisely those which are multiples of 3 (as discussed in the
Remark following Theorem 24 o
Ma 5c, Spring 2012
Caltech
Problem Set 3
Solutions
[13.5.11] Let F be the algebraic closure of F . Suppose f has no repeated irreducible
factors. We can write f = c fi with fi irreducible monic polynomials over F . Since F
is a perfect eld, F is separable
Ma 5c, Spring 2012
Caltech
Problem Set 4
Solutions
[14.1.10] Dene f : Aut(K/F ) Aut(K /F ) via 1 . Since and are both
isomorphisms, f ( ) is an automorphism of K . Take x F , then f ( )(x) = 1 (x) =
1 (x) = x, since 1 (x) F , and xes F . Therefore f is we
Caltech
Ma 5c, Spring 2012
Problem Set 5
Solutions
[14.3.8] Let be a root of f (x) = xp x a, then + k is clearly also a root of f (x)
for k Fp . Since f is of degree p, we get all the roots this way, and the splitting eld is
given by K = Fp []. Also, f ha
Caltech
Ma 5c, Spring 2012
Problem Set 6
Solutions
[14.6.13] a) We know that f (x) = x4 + ax2 + b = (x )(x + )(x )(x + ), so if f is
reducible it must factor as two quadratics. Since x must occur in a quadratic factor,
we distinguish three possible cases
Caltech
Ma 5c, Spring 2012
Problem Set 7
Solutions
[14.6.45] Consider D = 4p3 27q 2 .
Since f is irreducible, it is separable so D = 0 (recall that D = i<j (i j )2 ).
Now, since x3 1, 0, 1( mod 9) we get D 4p3 4, 0, 4( mod 9), thus D = 1.
[14.6.49] First
Ma 5c, Spring 2012
Caltech
Problem Set 8
Solutions
[14.2.22] a) By the denition of the norm we have that for K and G = Gal(K/F ):
NK/F ( ) =
( ) =
= NK/F ( ).
G
Thus if =
then NK/F () =
NK/F ( )
NK/F ( )
G
= 1.
b) Similarly, one has that T rK/F ( ) = T
Math 5c
Midterm
Due Wednesday, April 2, 12 pm
Do the following 5 problems in 3 hours. You may cite any theorems proven in class or Dummit and
Foote, but prove any material you use from other sources. You may use technology for calculations.
1. (15 points)
Math 5c
Midterm
Due Wednesday, April 2, 12 pm
Do the following 5 problems in 3 hours. You may cite any theorems proven in class or Dummit and
Foote, but prove any material you use from other sources.
1. (15 points) Let E be a eld extension of F . Proove o