課程名稱︰代數二
課程性質︰數學系大二必修
課程教師︰于靖
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/04/18
考試時限（分鐘）：180
試題 :
In answering the following problems, please give complete arguments as much
as possible. You may ask for any definition. For the first five problems, you
may use freely any Theorem already proved (or Lemmas, Propositions) from the
Course Lectures, or previous courses on Linear Algebra. Previous Exercises
assignments are also allowed to use in doing these five problems. You do not
need to give proofs of the theorems (statements) you are using, but you MUST
write down complete statements of the theorems which your arguments are based.
For the last problem, which is a Theorem from the Textbook. You are
requested to give direct proof, using only the definition of Tensor Product.
For your reference, formulas for quartic equations are give here:
4 3 2 4
f(x) := x + px + qx + r = Π (x - α ).
i=1 i
4 3 2 2 2 2 4 3
D = 16p r - 4p q - 128p r + 144pq r - 27q + 256r .
f
The resolvent cubic for the equation f(x) = 0 is:
3 2 2
h(x) = x - 2px + (p - 4r)x + q .
The roots of h(x) are: (α + α )(α + α ), (α + α )(α + α ), (α +
1 2 3 4 1 3 2 4 1
α )(α + α ).
4 2 3
＿1/2
1. Prove that |Q((2 + √2) ) is a cyclic Galois extension of |Q.
2. Let p ≠ 3, 5 be a prime number. Compute the Galois group of the polynomial
4
x + px + p over |Q.
3. Let K be a finite extension of |Q. Let f(x) ∈ K[x] be a separable
＿
polynomial. For each embedding σ of K into the field of algebraic numbers |Q,
σ
let f (x) be the polynomial obtained from f(x) by applying σ to the
coefficients of f(x). Then define the norm of f(x) to be
σ
N(f)(x) := Π f (x),
σ
＿
where σ runs through all the embeddings of K into |Q.
(1) Show that N(f)(x) ∈ |Q[x].
(2) Suppose that both f(x) and N(f)(x) are separable polynomials. Let N(f)(x) =
n
Π N (x) be the factorization into irreducible polynomials in |Q[x]. Prove that
i=1 i
n
f(x) = Π gcd(f(x), N (x)),
i=1 i
is a factorization of f(x) into irreducible factors inside K[x].
＿ ＿ ＿
4. Find a primitive generator for the field |Q(√2, √3, √5) over |Q. In other
＿ ＿ ＿
words, give an algebraic number α satisfying |Q(α) = |Q(√2, √3, √5).
＿ ＿ 1/2
5. Let α := ((2 + √2)(3 + √3)) , and E := |Q(α).
＿ ＿ ＿ ＿
(1) Show that a := (2 + √2)(3 + √3) is not a square in F := |Q(√2, √3).
＿ ＿
(Hint: Take norm N ＿ from F to |Q(√2). This will contradicts √6 is
F/|Q(√2)
＿
not an element of |Q(√2).)
(2) Prove [E: Q] = 8, and the roots of the minimal polynomial of α over |Q are
given by
＿ ＿ 1/2
±((2 ±√2)(3 ±√3)) .
＿ ＿ 1/2
(3) Let β := ((2 - √2)(3 + √3)) . Show that αβ ∈ F so that β ∈ E. Show
furthermore that E is Galois over |Q.
(4) Let σ ∈ Gal(E/|Q) be the automorphism sending α to β. Show that σ is an
element of order 4 in Gal(E/|Q).
＿ ＿ 1/2
(5) Let τ be the automorphism defined by τ(α) := ((2 + √2)(3 - √3)) .
Show that τ is also of order 4 in Gal(E/|Q). Show that τ and σ generate
2 2 3
Gal(E/|Q), σ = τ , and στ = τσ . In other words Gal(E/|Q) is the
quaternion group of order 8.
6. Let R be a commutative ring (with multiplicative identity 1). let M, M', N,
N' be R-modules. Give R-module homomorphisms φ: M → M', ψ: N → N'.
(1) Show that there exists a unique R-module homomorphism from M \otimes N to
R
M' \otimes N' sending m \otimes n to φ(m) \otimes ψ(n) for all m ∈ M and n
R
∈ N. This homomorphism is denoted by φ \otimes ψ in the following discussion.
(2) If λ: M' → M", and μ: N' → N" are also R-module homomorphisms, prove
that
(λ \otimes μ)。(φ \otimes ψ) = (λ。φ) \otimes (μ。ψ).