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 som
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
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
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) < (
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
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
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 =
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
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.
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
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 polyno
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
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,
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
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
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
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.
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.