(a) u a, b, c, d ∈ Q, J a + b√ 2 + c√ 3 + d√ 6 ∈ Q(√ 2,√ 3) (b) J [Q(√ 2,√ 3

(1)

Chapter 1. Field Extensions

(1) ' Q(α), Q(β), Q(γ) / Q Ý simple extensions Í α, β, γ 5½

α²− 2 = 0, β²− 3 = 0|C γ²− 4γ + 2 = 0.

(a) J Q(α)/Q õ Q(β)/Q Î isomorphic extensions over Q.

(b) J Q(α)/Q õ Q(γ)/Q Î isomorphic extensions over Q.

(c) uá α, γ ∈ R, J Q(α) = Q(γ).

(2) Ê R/Q Ý subextension Q(√ 2,√

3)/Q.

(a) u a, b, c, d ∈ Q, J a + b√ 2 + c√

3 + d√

6 ∈ Q(√ 2,√

3) (b) J [Q(√

2,√

3) : Q] = 4¬1 Q(√ 2,√

3)Ý-ôK°×

îW a + b√ 2 + c√

3 + d√

6 ÝP, Í a, b, c, d ∈ Q.

(c) u α = a + b√ 2 + c√

3 + d√

6 Í a, b, c, d ∈ Q v a, b, c, d / 0,

O a⁰, b⁰, c⁰, d⁰ ∈ Q ¸ÿ α⁻¹= a⁰+ b⁰√

2 + c⁰√

3 + d⁰√ 6.

(3) u K = K0 ⊆ K₁ ⊆ · · · ⊆ K_n−1⊆ K_n= L Î×=Ý field extensions. J [L : K] = [Kn : K_n−1] · · · [K₁ : K₀]vJ L/K Î finite extension uv°uEXb 1 ≤ i ≤ n, Ki/K_i−1 / finite extension.

(4) ' L/K Î×Í finite extension v p(x) ∈ K[x] Î×Í K[x] Ý

(2)

2 Exercise

(6) ' α Î algebraic over Q v α α²− 3α + 1 = 0. 0Õ β ∈ Q(α)

 β² = α.

(7) ' α Î transcendental over K, J3 K(α) 0Õ β ¸ÿ β² = α.

(8) ' L/K Î×Í field extension v α, β ∈ L, Í α algebraic over K,

 β transcendental over K. J α + β, α − β, αβ |C αβ⁻¹ / transcendental over K.

(9) ' K ⊆ F ⊆ L v F/K Î×Í algebraic extension. á α ∈ L, J

 α transcendental over K uv°u α transcendental over F .

———————————– 09 October, 2007

(3)

Chapter 2. Galois Group and Fixed Field

(1) ' α ∈ R α⁵= 7, L = Q(α). ' F L/Q Ý intermediate fields XW/), G Gal(L/Q) Ý subgroups XW/).

(a) O Gal(L/Q).

(b) O F.

(c) O G.

(d) u G : F → G, ¸ÿE F ∈ F L G(F ) = Gal(L/F ). ® G ÎÍ ×E×? ÎÍ ÌW?

(2)  L = Q(√ 2,√

3). ' F L/Q Ý intermediate fields XW/), G Gal(L/Q)Ý subgroups XW/).

(a) O Gal(L/Q).

(b) O F.

(c) O G.

(d) u G : F → G, ¸ÿE F ∈ F L G(F ) = Gal(L/F ). ® G ÎÍ ×E×? ÎÍ ÌW?

———————————– 16 October, 2007

(3) ' L/Q field extension. ' F L/Q Ý intermediate fields XW/

), G Gal(L/Q) Ý subgroups XW/). ' F : G → F, ÍL E H ∈ G F(H) = L^H = {λ ∈ L | σ(λ) = λ, ∀ σ ∈ H}.

(a) u L = Q(α) Í α ∈ R α⁵ = 7,EN×Í H ∈ G O F(H)

¬1 F ÎÍ ×E×? ÎÍ ÌW?

(b) u L = Q(√ 2,√

3),EN×Í H ∈ G O F(H) ¬1 F ÎÍ

×E×? ÎÍ ÌW?

(4)

4 Exercise

(4) Ê C = R(i), Í i ∈ C i² = −1. ' σ1, σ₂ ∈ Gal(C/R)

σ₁(i) = i|C σ2(i) = −i.

(a) 1 ¢ σ2(i) = −σ₁(i), ¬ σ2 6= −σ₁.

(b)  f = c1σ₁+ c₂σ₂,Í c1, c₂ ∈ C. uá f(1) = f(i) = 0, J

c₁= c₂= 0, ¬µh1 σ1, σ₂ linearly independent over C.

(5) ' L = Q(√ 2,√

3) v σi,j ∈ Gal(L/Q)  σij(√

2) = (−1)ⁱ√ 2 v σ_i,j(√

3) = (−1)^j√

3,Í 1 ≤ i, j ≤ 2.

(a) Ju

f = X

1≤i,j≤2

c_ijσ_ij v f(1) = f(√

2) = f (√

3) = f (√ 6) = 0, JEXb 1 ≤ i, j ≤ 2, /b cij = 0

(b) J

det







σ₁₁(1) σ₁₂(1) σ₂₁(1) σ₂₂(1) σ₁₁(√

2) σ₁₂(√

2) σ₂₁(√

2) σ₂₂(√ 2) σ₁₁(√

3) σ₁₂(√

3) σ₂₁(√

3) σ₂₂(√ 3) σ₁₁(√

6) σ₁₂(√

6) σ₂₁(√

6) σ₂₂(√ 6)





 6= 0

(6) ' L/K Î×Í finite extension v |Gal(L/K)| = [L : K] = n.  Gal(L/K) = {σ₁, . . . , σ_n}v α1, . . . , α_n∈ LÎ L/K Ý×à basis. J

det







σ₁(α₁) σ₂(α₁) · · · σ_n(α₁) σ₁(α₂) σ₂(α₂) · · · σ_n(α₂)

... ... . .. ... σ₁(α_n) σ₂(α_n) · · · σ_n(α_n)





6= 0.

(7) ' L/K Î×Í finite extension, F Î L/K Ý×Í intermediate field.

u σ ∈ Gal(L/K) Ê σ(F ) = {σ(λ) | λ ∈ F }.

(a) J σ(F ) ôÎ L/K Ý intermediate field.

(b) u α1, . . . , α_nÎ F/K Ý×à basis, J σ(α1), . . . , σ(α_n)Î σ(F )/K Ý×à basis, ¬µh1 [F : K] = [σ(F ) : K].

———————————– 30 October, 2007

(5)

(8) ' L = Q(√ 2,√

3)v σ ∈ Gal(L/Q) σ(√

2) = (−1)√

2v σ(√ 3) =

−√

3.  H = {1, σ} ⊆ Gal(L/Q), á F(H) = Q(√

6), ¿àJ

 |H| = [L : F(H)] Ý]°0Õ c1, c₂, c₃ ∈ Q(√

6) 0 v

c₁+ c₂√

2 + c₃√ 3 = 0.

Chapter 3. Normal Extension õ Separable Extension

(1) ' F/K Î×Í field extension, f(x) ∈ K[x] v L1, L₂ 5½Î f(x) over K õ f(x) over F Ý splitting field.

(a) u L/K Î×Í field extension L1 ⊆ L, 1 f(x) splits completely in L.

(b) u L/K Î×Í field extension K ⊆ L ⊆ L1 v f(x) splits completely in L, 1 L = L1.

(c) u L/K Î×Í field extension L1 ⊆ Lv L2⊆ L,J L1⊆ L₂. (d) J L1= L₂ uv°u F ⊆ L1.

(2) ' K Î×Í field, f(x) ∈ K[x] v L Î f(x) over K Ý splitting field.

u deg(f(x)) = n, J [L : K] | n!.

———————————– 6 November, 2007

(3) ' K Î×Í field v f(x) ∈ K[x]. ¿àó.hû°J f(x) over K Ý splitting field ÄD3.

(4) Ê f(x) = x²− 3. 3 R 0Õ f(x) over Q |C f(x) over Q(√ 2)Ý splitting field.

(5)  K = Q(√

2)v L = Q(√ 2,√

3). Ê σ ∈ Gal(K/Q) σ(√

2) = −√ 2.

(6)

6 Exercise

(6) ' L/K Î×Í finite extension v φ : L → L⁰ Î×Í ring isomorphism.

(a) J φ(K) = {φ(k) | k ∈ K} Î L⁰ Ý subfield.

(b) Ju α1, . . . , α_n∈ LÎ L over K Ý×à basis, J φ(α1), . . . , φ(α_n) Î L⁰ over φ(K)Ý×à basis, ¬µhÿ [L : K] = [L⁰ : φ(K)].

(7) ' L/K Î×Í field extension v [L : K] = 2, J L/K Î×Í normal extension.

(8) ' L/K Î×Í field extension v α1, . . . , α_n∈ L¸ÿ K(α1, . . . , α_n)/K Î×Í finite normal extension. ¨' F/K ôÎ×Í field extension v F ⊆ L,J F (α1, . . . , α_n)/F ôÎ×Í finite normal extension.

———————————– 20 November, 2007 (9) ' L/K Î field extension, f(x), g(x) ∈ K[x]. J f(x) | g(x) in K[x]

uv°u f(x) | g(x) in L[x].

(10) ' L/K Î×Í finite extension v charK = p. ' L Î separable over K,J p | [L : K].

(11) ' L = K(α) Î×Í finite extension, L ⊆ N v N/K Î×Í normal extension.

(a) ' charK = 0, J |MK(L, N )| = [L : K].

(b) ' charK = p v p - [L : K], J |MK(L, N )| = [L : K].

(c) ' charK = p v L/K Î separable extension, J

|M_K(L, N )| ≤ [L : K]

p .

———————————– 11 December, 2007

(7)

Chapter 4. Galois Extension

(1) ¾\¬1ìøÍ filed extension L/K Î Galois extension.

(a) K = Q, L = K(α) Í α Î x³− 2 = 0q.

(b) K = Q(ω), L = K(α) Í ω, α 5½Î x²+ x + 1 = 0 x³− 2 = 0

q.

(c) K = Q, L = K(ω, α) Í ω, α 5½Î x²+ x + 1 = 0 x³− 2 = 0

q.

(d) K = F₃(γ), L = K(β) Í γ Î transcendental over F3 v β Î x³− γ = 0q.

(e) K = F₃(γ), L = K(λ) Í γ Î transcendental over F3 v λ Î x⁴− γ = 0q.

(f) K = F₃(γ)(ζ), L = K(λ) Í γ Î transcendental over F3, ζ Î x²+ 1 = 0qv λ Î x⁴− γ = 0q.

(2) ¾\¬1ìøÍ filed extension L/K Î Galois extension.

(a) K Î×Í field v charK 6= 2, L = K(α) Í α over K Ý minimal polynomial degree 2.

(b) K = Q, L = K(β) Í β over K Ý minimal polynomial degree 3 vbÌq.

(8)

8 Exercise

(3)  L = Q(√ 2,√

3)C K = Q.

(a) J L/K Î Galois extension.

(b) 0 Gal(L/K) ÝXb subgroups.

(c) 0 L/K XbÝ intermediate fields.

(d) 0 Gal(L/K) ÝXbÝ normal subgroups.

(e) 0Xb L/K Ý intermediate field F ¸ÿ F/K Î Galois extension.

(4)  L = Q(α) Í α ∈ C Î x⁴+ x³+ x²+ x + 1 = 0qv K = Q.

(5)  L = Q(β, ω) Í β, ω ∈ C 5½Î x³− 2 = 0|C x²+ x + 1 = 0

qv K = Q.

———————————– 25 December, 2007

(a) u a, b, c, d ∈ Q, J a + b√ 2 + c√ 3 + d√ 6 ∈ Q(√ 2,√ 3) (b) J [Q(√ 2,√ 3

(a) u a, b, c, d ∈ Q, J a + b√ 2 + c√ 3 + d√ 6 ∈ Q(√ 2,√ 3) (b) J [Q(√ 2,√ 3