1. Let $K/F$ be a splitting field of a separable irreducible polynomial $f∈F[t]$.
    Show that $\gal(K/F)$ acts transitively on the set of roots $V(f)$ of $f$ in $K$.
    $∀α,β∈V(f)$, by 4.12 ∃ $F$-linear automorphism $ϕ\colon F(α)→F(β),α↦β$
    $K$ is splitting field of $f$ over $F(α)$.
    $K$ is splitting field of $ϕ(f)=f$ over $F(β)$.
    By 4.13 extend $ϕ$ to an automorphism $\barϕ\colon K→K$, then $\barϕ(α)=ϕ(α)=β$ and $\barϕ$ fixes $F$ as $ϕ$ does \begin{CD} K@>\barϕ>>K \\ @AAA@AAA\\ F(α)@>ϕ>>F(β)\\ @AAA@AAA\\ F@>\text{id}>>F \end{CD} Another proof: $∀α∈V(f)$ we need to prove orbit of $α$ is all of $V(f)$. The orbit polynomial $g(x)=\prod_{β∈G⋅α}(x-β)∈K^G[x]=F[x]$ annihilates $α$, so $f|g$, so$$\#V(f)=\deg f≤\deg g=\#(G⋅α)$$but $G⋅α⊆V(f)$, so $G⋅α=V(f)$.

  2. Let $f=t^3-s_1 t^2+s_2 t-s_3 ∈ F[t]$ have roots $α, β, γ$. Write $$ σ_i≔α^i+β^i+γ^i   \text{ for }   i ≥ 0 . $$
  3. Galois group of a biquadratic quartic
    Prove that $f≔t^4-4 t^2+2$ is irreducible over ℚ. Let $α ∈ ℂ$ be a root of $f$. Show that $ℚ(α)$ is the splitting field of $f$. Prove that $\gal(ℚ(α)/ℚ)$ is cyclic of order 4.
    By Eisenstein with $p=2$, $f$ is irreducible over ℚ. $$t^4-4t^2 + 2 = 0 ⇔ t^2 =2 ± \sqrt{2}$$ so the roots of $f$ are $±α,±β$ where $α=\sqrt{2+\sqrt2},β=\sqrt{2-\sqrt2}$ $$αβ=\sqrt2=α^2-2⇒β∈ℚ(α)$$ so ℚ(α) is the splitting field of $f$ over ℚ, so $[ℚ(α):ℚ] = 4$, so $\gal(ℚ(α)/ℚ)=C_2^2$ or $C_4$.
    Consider $σ∈\gal(ℚ(α)/ℚ);σ(α)=β$ \begin{array}l σ^2(α)=σ(β)=σ\left(\frac{α^2-2}α\right)=\frac{β^2-2}β=\frac{-\sqrt2}β=-α\\σ^3(α)=σ(-α)=-σ(α)=-β\\σ^4(α)=σ(-β)=-σ(β)=α\end{array} so σ has order 4, but no element of $C_2^2$ has order $4$. So $\gal(f) ≅ C_4$.
    Remark: $α=2\cos(π/8)$, so $ℚ(α)$ is a subextension of cyclotomic extension $ℚ(ζ_8)$ with Galois group $ℤ^×_8≅C_4×C_2$.
  4. Let $K/F$ be a finite extension. Recall that $K/F$ is said to be separable if $m_{F, α}$ is a separable polynomial for all $α ∈ K$. Prove that $K/F$ is Galois if and only if it is normal and separable.

    (⇒) Given a finite Galois extension $K/F$. By Theorem 6.6, $K/F$ is normal.

    By Corollary 6.7a, $K/F$ is separable.

    (⇐) Given a finite, normal, separable extension $K/F$. Let $α_1,α_2,…,α_n$ be a $F$-basis for $K$.

    Since $m_{F,α_i}$ has a root $α_i$ in $K$, by normality, $m_{F,α_i}$ splits over $K$.

    Irreducible factors of $f≔\prod_{i=1}^nm_{F,α_i}$ are $m_{F,α_i}$ which are separable since $K/F$ is separable. Thus, $f$ is separable.
    Since $α_1,α_2,…,α_n$ generate $K$, $K$ is the splitting field of $f$, then $K/F$ is Galois.

  5. Suppose that $\operatorname{char} F ≠ 2$. Let $f=t^4+p x^2+q t+r$ have roots $α_1, α_2, α_3, α_4$ and let $$ v_1≔α_1 α_2+α_3 α_4,   v_2≔α_1 α_3+α_2 α_4,   v_3≔α_1 α_4+α_2 α_3 $$ Show that $(t-v_1)(t-v_2)(t-v_3)=t^3-p t^2-4 r t+(4 p r-q^2)$.
    By Vieta’s formulas \begin{gather*} 0=α_1+α_2+α_3+α_4 \\p=α_1 α_2+α_1 α_3+α_1 α_4+α_2 α_3+α_2 α_4+α_3 α_4 \\-q=α_1 α_2α_3+α_1α_2 α_4+α_1α_2 α_4+α_2α_3 α_4 \\r=α_1 α_2α_3α_4 \end{gather*} we calculate symmetric polynomials of $v_1,v_2,v_3$ \begin{gather*} v_1 + v_2 + v_3 = p,\\ \begin{split} v_1 v_2 + v_1 v_3 + v_2 v_3 &= (α_1 + α_2 + α_3 + α_4) (α_1 α_2 α_3 + ⋯ + α_2 α_3 α_4) - 4α_1 α_2 α_3 α_4 \\ &= 0(-q) - 4r, \end{split} \\ \begin{split} v_1 v_2 v_3 &= (α_1 α_2 α_3 + ⋯ )^2 + α_1 α_2 α_3 α_4 \bigl\{(α_1 + ⋯ )^2 - 4( α_1 α_2 + ⋯) \bigr\} \\ &= (-q)^2 + r(0^2 - 4p). \end{split} \end{gather*} Hence $v_1,v_2,v_3$ are the roots of $t^3-p t^2-4 r t+(4 p r-q^2)$.
    Another method: Let $v=α_1α_2+α_3α_4$, $w=α_1α_2-α_3α_4$, $τ=α_1+α_2$.
    Since $α_1+α_2+α_3+α_4=0$ then $-τ=α_3+α_4$.
    Also, $\frac{v+w}2=α_1α_2,\frac{v-w}2=α_3α_4$. \begin{align*} t^4+pt^2+qt+r&=(t-α_1)(t-α_2)(t-α_3)(t-α_4)\\&=(t^2-(α_1+α_2)t+α_1α_2)(t^2-(α_3+α_4)t+α_3α_4)\\&=(t^2-τ t+\frac{v+w}2)(t^2+τ t+\frac{v-w}2)\\&=t^4+(v-τ^2)t^2+τ wt+\frac{v^2-w^2}4 \end{align*} Equating coefficients,\[p=v-τ^2,q=τ w,4r=v^2-w^2\] Expressing $w^2,τ^2$ in $v,r,p$ $$4r=v^2-w^2⇒w^2=v^2-4r$$ $$p=v-τ^2⇒τ^2=v-p$$ then$$q^2=τ^2w^2=(v-p)(v^2-4r)=v^3-pv^2-4rv+4rp$$ Hence $v$ is a root of $t^3-pt^2-4rt+(4pr-q^2)$.
  6. Let $K/L$ and $L/F$ be finite separable extensions.