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Need someone who knows abstract algebra well. Need 1 and 3 done of the assignment in the attached file.Show all your work.
MAT401 Polynomial Equations and Fields
Assignment 5
Due Wednesday August 3 at the beginning of the lecture Please write your arguments neatly and clearly. Numbers in [ ] indicate how much a question
or a part of it is worth. The assignment is out of 50. Throughout, the letters F, F 0 , K and L denote
fields.
1. [7] Determine if each statement is true or false. No explanation is necessary. (But make sure you
know exactly why a given statement is true or false.)
We use the following notation: If α is algebraic over F, the minimal polynomial of α over F is
denoted by mα,F (x).
(a) Every subfield of C contains Q.
(b) There are no ring homomorphisms Q → Z.
(c) If F and F 0 are finite extensions of Q such that [F : Q] = [F 0 : Q], then every ring homomorphism F → F 0 is actually an isomorphism.
(d) If F and F 0 are finite extensions of Q such that [F : Q] = [F 0 : Q], then F and F 0 are isomorphic
as rings. (e) If F ⊂ K, α ∈ K is algebraic over F, and f(x) ∈ F[x] is such that f(α) = 0, then mα,F (x) f(x).
(f) If F ⊂ K, α ∈ K is a root of f(x) ∈ F[x] of degree n ≥ 1, then [F(α) : F] ≤ n.
(g) If Q ⊂ F ⊂ C and F/Q is finite, then there is a polynomial f(x) ∈ Q[x] such that F is
contained in the splitting field of f(x) over Q. (h) If F ⊂ K ⊂ L, and α ∈ L is algebraic over F, then mα,F (x) mα,K (x) in K[x].
(i) If F ⊂ K ⊂ L, and α ∈ L is algebraic over F, then mα,K (x) mα,F (x) in K[x].
(j) The polynomial x8 + 6x3 + 9x + 21 has 8 distinct roots in C.
(k) If F ⊂ K and α ∈ K is such that α3 is algebraic over F, then α is also algebraic over F.
(l) Every algebraic extension is finite.
(m) If F ⊂ C, then every ring homomorphism F → C fixes Q.
(n) If f(x) ∈ Q[x] is irreducible over Q and has degree n, and α1 , · · · , αn ∈ C are the roots
of f(x), then every ring homomorphism ϕ : C → C restricts to an automorphism of
Q(α1 , · · · , αn ). (In other words, the statement is claiming that if ϕ : C → C is a ring
homomorphism, then the association z 7→ ϕ(z) defines an isomorphism Q(α1 , · · · , αn ) →
Q(α1 , · · · , αn ).)
2. [6] (a) [2] Suppose K/F is a field extension, α ∈ K such that α2 ∈ F. Show that [F(α) : F] is either
1 or 2.
√
/ Q(α1 , · · · , αn ).
(b) [4] Suppose α1 , · · · , αn ∈ C are such that α2i ∈ Q for each i. Show that 5 2 ∈
1 2 3. [12] Let us give a definition first. We say a finite extension K/F is simple if there is ω ∈ K such
that K = F(ω). The goal of this question is to prove the following theorem: If F ⊂ K ⊂ C and K/F
is finite, then K/F is simple.
(a) [1] Argue that to prove the theorem it suffices to prove the following: If F ⊂ C and
α1 , · · · , αn ∈ C are algebraic over F, then F(α1 , · · · , αn ) is a simple extension of F.
(b) [9] On page 4 of this document, you can find a sketch of a proof of the statement given
in Part (a), with several steps and justifications left to you. Fill in the “gaps”. (You don’t
have to rewrite the full argument; just complete the gaps. If you decide to rewrite the
completed argument,
please clearly indicate where you address each gap.)
√
(c) [2] Let K = Q( 2, i). Following the method given in the proof of (b), find ω such that
K = Q(ω).
4. [6] Suppose ϕ : R → S is a ring isomorphism (i.e. a bijective ring homomorphism).
(a) [3] Show that ϕ−1 : S → R is also a ring isomorphism.
(b) [3] Show that R is an integral domain if and only if S is an integral domain
5. [8] Let L be the splitting field of xn − 3 over Q. Let ζn = e2πi/n and α = √
n 3. (a) [2] Show that L = Q(α, ζn ).
(b) [3] Suppose for the rest of the question that n = p is prime. Find [L : Q].
(c) [3] Show that f(x) = 1 + x + x2 + · · · + xp−1 is irreducible over Q(α).
6. [11] Let us start with a definition. Given F ⊂ K ⊂ C with K/F finite, we say the extension K/F
is Galois if it satisfies any (and hence all) of the equivalent conditions of Theorem 5 of the notes.
If K/F is Galois, then we call the group Aut(K/F) the Galois group of K/F. It is constumary to use
the notation Gal(K/F) for Aut(K/F) in this case. (So Gal(K/F) and Aut(K/F) are the same thing,
except that we use the first notation only if K/F is a Galois extension.)
Below all fields are subfields of C.
(a) [1] Let K be a Galois extension of F. Let f(x) ∈ F[x] be an irreducible polynomial which has
a root in K. Is it true that K contains all (complex) roots of f(x)? No explanation necessary.
(b) [4] Let K/F be Galois, and that f(x) ∈ F[x] be a nonzero polynomial all whose complex
roots are in K. Let α1 , · · · , αn be all the distinct roots of f(x), and for brevity denote the set
{α1 , · · · , αn } by roots(f(x)). Let σ ∈ Gal(K/F). Show that there is a bijection
roots(f(x)) → roots(f(x)) (1) given by αi 7→ σ(αi ). (In other words, show that σ permutes the roots of f(x).) Denote the
bijection above by σ roots(f(x)) , the restriction of σ to the set of roots of f(x).
(c) [2] For any nonempty set X, denote the symmetric group on X by SX . Recall that as a set
SX is the set of all bijections X → X, and the group operation is composition of functions.
Continuing with the notation as in (b), show that Gal(K/F) → Sroots(f(x))
σ 7→ σ roots(f(x)) is a group homomorphism. (One may refer to this map as the restriction to the set of roots
of f(x).) 3 (d) [2] Now suppose moreover that K is the splitting field of f(x) over F. Show that the map
(1) above is injective.
We usually identify Gal(K/F) with its image under this injection, and think of an element of the Galois group as a permutation of the roots of f(x).
(e) [1] Read the example on page 46 of the notes (done in class on Wednesday July 27). With
the notation as in the example, identifying the Galois group Gal(L/Q) with a subgroup of
S{α1 ,α2 ,α3 } , is complex conjugation equal to the transposition (α2 α3 )?†
(f) [1] Agian in regards to the example on page 46 of the notes, show that Gal(L/Q) ' S3 . † For any finite nonempty set X, the cycle notation in SX is just like the case of Sn = S{1,··· ,n} . Here (α2 α3 ) refers to
the element of S{α1 ,α2 ,α3 } that sends α2 7→ α3 , α3 7→ α2 , and α1 7→ α1 . 4 T HEOREM 1. If F ⊂ C and α1 , · · · , αn ∈ C are algebraic over F, then F(α1 , · · · , αn )/F is a simple
extension.
P ROOF. First note that the result certainly holds when n = 1: F(α1 )/F is finite (as α1 is algebraic
over F) and is clearly simple. Thus we need to prove the result for n ≥ 2. We do this by induction
on n. Let us assume the base case (n = 2) for the moment.
Gap 1: Carry out the induction. In other words, suppose the result holds for some n ≥ 2, and
prove it for n + 1. (Note that we are assuming the base case for now. You can use it.)
Now we turn our attention to the base case, i.e. when n = 2. Suppose α, β ∈ C are algebraic over
F. Our goal is to show that there is ω such that F(α, β) = F(ω). Let f(x) (resp. g(x)) be the minimal
polynomial of α (resp. β) over F. Let k = deg(f(x)) and l = deg(g(x)). Let α1 = α, α2 , · · · , αk be
the roots of f(x) in C, and β1 = β, β2 , · · · , βl be the roots of g(x) in C. Let c ∈ F be an element that
is not equal to any of the numbers
αi − α
(1 ≤ i ≤ k, 2 ≤ j ≤ l).
β − βj
Gap 2: How de we know such c exists?
Set ω = α + cβ. We claim that F(α, β) = F(ω).
Gap 3: Is F(ω) ⊂ F(α, β)? Why?
To establish the claim, we need to show that F(α, β) ⊂ F(ω). For this it suffices to show α, β ∈ F(ω).
Define h(x) := f(ω − cx).
Gap 4: Is h(x) ∈ (F(ω))[x]? Why?
Gap 5: Verify that β a root of h(x).
Gap 6: Show that none of β2 , · · · , βl can be a root of h(x).
Let g1 (x) be the minimal polynomial of β over F(ω). Note that in particular, g1 (x) ∈ (F(ω))[x]. Gap 7: Does it follow that g1 (x) g(x) and g1 (x) h(x)? Why?
Gap 8: Argue that β is the only root of g1 (x) (in C).
Thus g1 (x) is of the form a(x − β)r for some r ≥ 1 and a ∈ F(ω). Since g1 (x) is monic, a = 1, and
g1 (x) = (x − β)r . Since g1 (x) is irreducible over F(ω), it cannot have any repeated roots. This r = 1
and g1 (x) = x − β.
Gap 9: Does it follows that β ∈ F(ω)? Why?
Gap 10: Use β ∈ F(ω) and ω = α + cβ to conclude that α ∈ F(ω) as well. This completes the
proof of the theorem.
5 Practice Problems. The following problems are for your own practice. Please do not hand them
in. Throughout F and K denote fields.
1. Suppose ϕ : R → S is a ring isomorphism. Let a ∈ R. Show that a ∈ U(R) if and only if
ϕ(a) ∈ U(S).
2. Suppose R and S are isomorphic rings. Show that R is a field if and only if S is a field.
3. Let L be the splitting field of x3 + 9x + 3 over Q. Show that [L : Q] = 6 and that Gal(L/Q) ' S3 .
4. Suppose ϕ : R → S is a ring isomorphism. Let a ∈ R. Show that a is irreducible in R if and only
if ϕ(a) is irreducible in S.
5. Suppose ϕ : R → S is a ring isomorphism. Let I ⊂ R be an ideal. Show that I is a prime ideal if
and only if ϕ(I) is a prime ideal of S.
6. Suppose F ⊂ K ⊂ C and [K : F] = 2. Show that K/F is a Galois extension.
More practice problems to be posted.