CORRECTIONS

ALGEBRA BY T.T.MOH

CHAPTER I

Set Theory and Number Theory

Page 4, line 18: (b) for n ≥ 2, the truth of P(ℓ) for all ℓ < n implies the truth of P(n). Page 35, line 26: ⋯, ∀ a ∈ ℚ\{0}. Page 38, line 14: · · ·. Let a ≠ b, D(a, b) = ϵ. Page 39, line 12:

\begin{matrix}a − tp^{ℓ} = p^{ℓ}(\frac{m}{n} − t) = p^{ℓ}(\frac{m − nt}{n}) = p^{ℓ+ℓ′} \frac{m′}{n} , p ∤ m′ \\ D_{p}(a, tp^{ℓ}) = qp^{−ℓ−ℓ′} < p^{−ℓ} \end{matrix}

Page 39, line 15: ⋯ we find a power series expansion of a = ∑_{i=ℓ}^{∞} t_{i}p^{i} Page 39, line 20:

D(−\frac{1}{6}, 3^{−1} + 1 + 3 + 3^{2} + ⋯ + 3^{n}) = 3^{−n−1} → 0

CHAPTER II

Group Theory

Page 48, line 22: ⋯ which satisfies a^{n} = e Page 59, lines 26-28:

η_{a}(x, y) = ((x + a)′, (y + ca)′), ∀ (x, y) ∈ T_{2}

where (x+a)′ is the unique real number which satisfies 0 ≤ (x+a)′ < 1, (x+a)−(x+a)′ ∈ ℤ. Page 60, line 13: Find Orb(B) for all B ∈ O(n, ℝ). Page 60, line 17: (11) Let the group O(n, ℝ) act on the set Symm(n, ℝ) of all real

symmetric matrices Page 60, line 19:

A(S) = ASA^{−1}, where A ∈ O(n, ℝ), T ∈ Symm(n, ℝ)

Page 63, line 31:

θ = nθ_{0} + r, 0 ≤ r < θ_{0}

Page 67, line 2: g_{1}(−1) ∗ g ∗ g_{1} is obviously surjective. · · · Page 67, line 8:

(1)  2 − \frac{2}{n} = ∑_{orbits}(1 − \frac{v_{s}}{n}) = ∑_{orbits}(1 − \frac{1}{n_{s}})

Page 67, line 10:

n_{s} = o(Stab(s)) ≥ 2

Page 70, line 16: Similar to part (5), let us view s_{3} as the north pole. Since v_{s_{3}} = 12, then there are 12

Page 71, lines 18-19:

G = \{x ↦ \frac{ax + b}{cx + d} : det ( \begin{matrix}a, & b \\ c, & d \end{matrix} ) ≠ 0\}

Let T= {0, 1, ∞}. Find Stab(T).

Page 71, line 21: (ℤ × ℤ). Page 72, line 10: Card(class) ≤ Card(G^{(1)}) for any conjugate class of G.

Page 74, line 30:

≠ (0, 2) = ρ_{90}((2, 0)) = ρ_{90} ∗ β_{(1,0)}((1, 0))

Page 75, line 15: (2, 5)(3, 4). page 78, line 4: g_{1}, g_{2} ∈ G with Page 78, line 9: Moreover, e′ = ρ(e) ∈ H’, H’≠ ∅, therefore, H’ is a subgroup. Page 78, lines 11-12: F ≠ ∅. Let g_{1}, g_{2} ∈ F. Then we have

ρ(g_{1} ∗ g_{2}^{−1}) = ρ(g_{1}) ∗ ρ(g_{2}^{−1}) = ρ(g_{1}) ∗ ρ(g_{2})^{−1} ∈ F’

Page 79, line 26: ⋯ subgroup of G’ and H the pre-image of H’. The induced Page 82, line 18:

g_{1}^{−1} ∗ g_{2}(s) = h(s) = s

Page 83, line 29: see that σ(1) = [m]_{n} must generate ℤ_{n}. Therefore we must have the following with

Page 83, line 34: Aut(ℤ_{n}) and π(σ) = [m]_{n}. page 84, line 17: Proof Let g ∈ G, g_{1} ∈ Inn(G) and σ ∈ Aut(G). Then we have Page 84, line 23: (2) Prove that Aut(ℚ)=ℚ^{∗}. Page 86, line 5: normal subgroup G_{1}. Let G^{∗} be G/G_{1}. Then o(G^{∗} = p^{n−1} and ⋯ Page 87, line 11: It is clear that o(H)=Card(H(a))≤ Card(A)= p^{n}. Page 87, line 21:

g ∗ H ∗ g^{−1} = Stab(g∗H), ∀ g ∈ G

Page 100, line 28: (7) Let p be a prime factor of a positive integer n with 2 < p < n. ⋯

CHAPTER III

Polynomials

Page 102, line 16: (3) Distributive law: for any a, b, c ∈ K, we have Page 103, line 5: distributive law, we have Page 104, line 24:

u_{3} = c_{1} + c_{3} + c_{4} + c_{7}

Page 105, line 21: and there is the multiplicative unity 1, namely 1 · a = a · 1 = a for all element a ∈ R.

Page 105, line 22: (3) Distributive law: for any a, b, c ∈ R, we have Page 105, lines 29-30: that it is not required the existence of the multiplicative unity

1. In this case the ring in our definition will be called a ring with unity. We shall use the term ring without unity

Page 106, line 2: existence of multiplicative unity. For instance, 2ℤ is a ring without unity.

Page 107, line 1: (4) Prove that FL(n, ℝ) is not a commutative ring. Page 107, line 22: ⋯ associative and if there is a unity. Let R be a ring and Page 108, lines 20-21: (1) It is easy to see that R[x] is a ring. The unity for summation

0 for R is the unity for summation for R[x]. The unity for multiplication 1 is the unity for multiplication for R[x].

Page 108, line 22: (2) Let ℤ_{+} be the semi-group (Cf Exercise 10, §1) of all ⋯ Page 111, lines 20-21:Proof It follows from the preceding theorem that if f(x), g(x) are

non-zero, then deg(f(x) g(x)) ≥ 0, and hence f(x) g(x) ≠ 0.

Page 115, line 6:

g(x) = b_{0} + b_{1}x + b_{2}x^{2} + ⋯ + b_{m}x^{m}, b_{m} ≠ 0

Page 117, line 9: the rings K[x_{1}, x_{2}, ⋯, x_{n}]. For this purpose, let us introduce, Page 119, line 2:

α_{i}(x) ≡ 1 (mod (x − b_{i})), α_{i}(x) ≡ 0 (mod (x − b_{j})) ∀ j ≠ i

Page 119, line 10:

f(x) ≡ a_{i} (mod (x − b_{i})) ∀ i i.e, f(b_{i}) = a_{i}

Page 120, line 3: ⋯ then the relation α_{1} = δ_{2}α_{2} = δ_{2}δ_{1}α_{1} implies (1 − δ_{2}δ_{1})α_{1} = α_{1} implies δ_{2}δ_{1} = 1.

Page 121, lines 19-20:

\begin{matrix}β = ϵ_{2} ∏ π_{i}^{m_{i}}, γ = ϵ_{3} ∏ π_{i}^{n_{i}} \\ πγ = ϵ_{3}π ∏ π^{n_{i}} = αβ = ϵ_{1}ϵ_{2} ∏ π_{i}^{ℓ_{i}} ∏ π_{i}^{m_{i}} \end{matrix}

Page 121, line 27:

π_{1} | α = ϵχ_{1}(χ_{1}^{n_{1}−1} ∏_{j>1} χ_{j}^{n_{j}})

Page 121, line 29:

π_{1} | χ_{1} or π_{1} | (χ_{1}^{n_{1}−1} ∏_{j>1} χ_{j}^{n_{j}})

Page 122, lines 1-2: If π_{1} ∤ χ_{1}, then we have π_{1} | χ_{2}(χ_{2}^{n_{2}−1} ∏_{j>2} χ_{j}^{n_{j}}), namely

π_{1} | χ_{2} or π_{1} | (∏_{j>2} χ_{j}^{n_{j}})

Page 122, line 14: α_{0}, α_{1}, ⋯, α_{n} of f(x). If deg(f(x))≥ 1 and c(f(x)) is the set of all units, then ⋯

Page 124, lines 1-2:

Theorem 3.9. Let D be an integral domain such that every element can be written as a product of irreducible elements. Then every non-zero polynomial f(x) ∈ D[x] can be written as a product of irreducible elements in D[x].

Page 126, line 23: Theorem 3.7., it suffices to show that every irreducible element in D[x_{n}] is prime. · · ·

Page 129, line 6: ⋯ a_{i} is nilpotent, namely there is m_{i} such that a_{i}^{m_{i}} = 0, for all i > 0. Page 129, line 12-14: (6) Let a_{1}, a_{2}, ⋯, a_{n} be distinct integers. Prove the following

polynomial f(x) is irreducible in ℚ[x],

f(x) = (∏_{i=1}^{n}(x − a_{i})^{2}) + 1

Page 144, line 19:

(A) = \{ \begin{matrix}\{0\}, & ifA = ∅ \\ \{∑_{finite} r_{i}a_{i} : r_{i} ∈ R, a_{i} ∈ A\}, & otherwise \end{matrix}

Page 144, line 23: that K= ℚ, or ℝ, or ℂ. Let A = {f_{i}(x_{1}, ⋯, x_{n})} ⊂ K[x_{1}, ⋯, x_{n}] be ⋯

Page 146, line 17: (3) ρ(1) = 1′, where 1 and 1′ are unities of R and R’ respectively. Page 147, line 6: (2) Let the map χ: R→ R/I be defined as χ(r) = [r]. Then χ ⋯ Page 147, lines 25-26:

\begin{matrix}= [ρ(r_{1} + r_{2})] = [ρ(r_{1}) + ρ(r_{2})] \\ = [ρ(r_{1})] + [ρ(r_{2})] \end{matrix}

Page 150, line 10: (1) In our definition of rings, we assume that there is a unity 1. Let a = 1 ⋯

Page 151, line 18: Theorem (Dedekind & Hasse) ⋯ Page 151, line 20: there exist u, v ∈ R with 0 < N(va − ub) < N(a), then R is a

P.I.D..

Page 151, lines 23-24: ⋯, then we have u, v ∈ R with 0 < N(va − ub) < N(a). Note that va − ub ∈ I. Therefore, it is impossible and a | b.

Page 151, line 26: Let a be any element in I \ {0} such that N(a) is minimal. · · · Page 151, line 29: ⋯ Note that the previous inequality is equivalence to N(u(b/a) −

v) < 1.

Page 151, line 33:

u(b/a) − v = (x + y\sqrt{−19})(s + r\sqrt{−19})/z − (q − t\sqrt{−19})

Page 152, line 2:

N(u(b/a) − v) = (ℓ^{2} + 19)/z^{2} < 1

Page 156, line 2:

d_{p}(p) = 1,  d_{i}(p) ≡ 0( mod p),  c_{i}(p) ∈ ℤ

Page 156, line 28:

f(x, y) = x^{2} + y^{2} − 1 = 0

Page 158, line 26: (ii) Let I be a proper ideal in C([a, b]). Then ⋯ Page 159, line 3: (15) Let D be a multiplicative set in ⋯ Page 159, line 5: (16) Let R be a Noetherian integral domain and D a multiplicative

set. · · ·

CHAPTER IV Linear Algebra

Page 173, lines 6 & 7:

\begin{aligned}L′(D) & = (D − a_{1})^{m_{1}} ∏_{j>1}(D − a_{j})^{m_{j+1}} \\ & = ∏_{j>1}(D − a_{j})^{m_{j+1}}(D − a_{1})^{m_{1}} \end{aligned}

Page 176, line 5:

subset R ⊂ S, R ⊃ W and a bijective map ρ : R → S′ with ⋯ Page 177, lines 17 & 19:

s′ ∉ [\overline{T}] = the subspace generated by \overline{T} .

[\overline{T}] ≠ V = [S]

Page 178, line 2:

(1)  as + ∑_{finite} a_{i}t_{i} + ∑_{finite} b_{j}t′_{j},  t_{i} ∈ \overline{T} , t′_{j} ∈ §′\\overline{T}′

CHAPTER V Polynomial in One Variable and Field Theory

Page 255, line 9:

lim_{x→∞} f(x) = ∞

Page 257, line 8: (4) Let f(x) = x^{n} + a_{1}x^{n−1} + ⋯ + a_{n−1}x + a_{n} ∈ ℤ[x] with a_{n} ≠ 0. Suppose ⋯

Page 257, line 13:

g(x) = ∑_{j=0}^{n} \binom{n}{j}a_{j}x^{n−j}

Page 257, line 25: a K-injection. If σ is an isomorphism. Then σ is called a Kisomorphism.

Page 259, line 14: Since K[α] ⊂L, then it is an integral domain. It follows ⋯ Page 264, line 30: △oab ∼ △cad. If we let \overline{oa} = α, \overline{ca} = 1, and cd = β, then we have

the product ⋯

Page 270, line 6:

[K(x) : K(\frac{f(x)}{g(x)})] = n

Page 270, line 16: (7) Prove that [ℚ[e^{2πi/8}] : ℚ[i]] = 2. Page 271, line 3: exists an integer n ≠ 0 such that nα is an algebraic integer. Page 272, line 8: Let f(x) be an irreducible element in K[x]. Assume L is an extension

of K such that there is a root α of f(x) in L. Let L’ be an Page 279, line 24: Therefore φ_{n}(x) ∈ ℤ[x]. Page 281, lines 29-30: (3) Let K be a finite field of 2^{n} elements. Show that a polynomial

f(x) ∈ ℤ_{2}[x] of degree n is primitive if and only if f(x) satisfies the following two Page 282, line 18: root of an irreducible polynomial f(x) ∈ ℤ_{p}[x] of degree m. · · · Page 282, line 26:

K ∩ ℝ = ℚ[e^{2πi/p} + e^{−2πi/p}]

Page 286, line 9:

KL^{p} = {∑_{i=1}^{n} k_{i}ℓ_{i}^{p} : k_{i} ∈ K, ℓ_{i} ∈ L.}

Page 291, line 13: (7) Let L⊃ R⊃ K be field extensions. Suppose that L is ⋯ Page 291, line 15: (8) Let L⊃ R⊃ K be field extensions. Suppose that L is ⋯ Page 304, line 2: of L. Let S be an extension of K in Ω . Then the composition, L·S,

is a Galois ⋯ Page 304, line 20: ⋯ group G(R/S) is a subgroup of the multiplicative group ℤ_{n}^{∗} = Page 307, line 20: Definition 5.19 If there exists a chain of radical extensions as

follows, Page 309, line 2: cyclic group of order n, then f(x) can be solved by radicals. Page 310, lines 3-8: Let R=K_{1}[β_{1}]. Then R is a radical extension of K_{1}.

(5) We claim R=K_{2}. Note then our theorem is established. It is clear that β_{1} ∈ K_{2}, and R⊂ K_{2}. On the other hand, none of the σ^{j}’s are the identity map on R, therefore, we have

m ≤ o(G(R/K_{1})) = [R: K_{1}] ≤ [K_{2} : K_{1}] = m

We conclude that R=K_{2}.

Page 313, line 22: It is easy to see that L=F(x_{1}, x_{2}, x_{3}), and the ⋯ page 315, line 11:

Δ ↦ \overline{Δ} = \sqrt{−4θ_{2}^{3} − 27θ_{3}^{2}}