Math 4120A/9020A Assignment 3
1. Let k be a field.
(a) Show that the mapping ϕ : k[t] → k[t] defined by ϕ(f (t)) =
f (at + b) for fixed a, b ∈ k, a 6= 0 is an automorphism of k[t] which
is the identity on k.
(b) Conversely, let ϕ be an automorphism of k[t] which is the identity
on k. Prove that there exist a, b ∈ k, a 6= 0 such that ϕ(f (t)) =
f (at + b) as in (a).
2. This exercise determines Aut(R/Q).
(a) Prove that any σ ∈ Aut(R/Q) takes squares to squares and takes
positive reals to positive reals. Conclude that a < b implies that
σ(a) < σ(b).
(b) Use (a) and the fact that Q is dense in R (more specifically for any
interval (a, b) there exists c ∈ (a, b)∩Q) to show that Aut(R/Q) =
{id}.
3. Prove that the automorphisms of the rational function field k(t) which
fix k are precisely the fractional linear transformations determined by
for a, b, c, d ∈ k, ad−bc 6= 0. (So f (t) ∈ k(t) maps to f ( at+b
).).
t → at+b
ct+d
ct+d
4. (Wedderburn’s Theorem on Finite Division Rings) This exercise outlines a proof (following Witt) of Wedderburn’s theorem that a finite
division ring D is a field (i.e. commutative).
(a) Let Z be the centre of D. Prove that Z is a field containing Fp
for some prime p. If Z = Fq , prove that D has order q n for some
n.
(b) The nonzero elements D× of D form a multiplicative group. For
any x ∈ D× show that the elements of D which commute with x
form a division ring which contains Z. Show this division ring has
order q m for some m ∈ N and that m < n if x 6∈ Z.
(a) Show that the class equation of the group D× is
q n − 1 = (q − 1) +
1
r
X
qn − 1
|CD× (xi ) |
i=1
where x1 , . . . , xr are representatives of the distinct conjugacy classes
in D× not contained in the centre of D× . Conclude from (b) that
for each i, |CD× (xi )| = q mi − 1 for some mi < n.
n
is an interger then mi divides n. Conclude
(d) Prove that since qqmi−1
−1
n
xn −1
that Φn (x) divides xmi −1 and hence that Φn (q) divides qqmi−1
for
−1
i = 1, . . . , r.
Q
(e) Prove that (c) and (d) imply that Φn (q) = ζ primitive (q − ζ)
divides q − 1. Prove that |q − ζ| > q − 1 for any root of unity
ζ 6= 1. Conclude that n = 1. i.e. that D is a field.
5. The construction of the regular 7-gon amounts to the constructibility of
cos(2π/7). Show that α = 2 cos(2π/7) satisfies the equation x3 + x2 −
2x − 1 = 0. Use this to prove that the regular 7-gon is not constructible
by straightedge and compass.
6. Show that α = 2 cos(2π/5) satisfies the equation x2 + x − 1 = 0 to
conclude that the regular 5-gon is constructible by straightedge and
compass.
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