Field and Galois Theory

$\sim$ is an equivalence relation.

If $G$ acts transitively, then all equivalence classes are distinct and contain the same number of elements.

If $\operatorname {Card}(X)$ is a prime number and if $G$ acts transitively and contains at least one transposition then $G$ must be the whole permutation group of $X$.

Let $A, B \in K[X]$, $AB \notin K$, $\gcd (A,B) = 1$, and put $y = A(x)/B(x)$. Show in succession that the polynomial $A - yB \in K(y)[X]$ is not $0$, that $x$ is algebraic over $K(y)$, that $y$ is transcendental over $K$, that $A - yB$ is irreducible in $K(y)[X]$, and that

Show that the elements of $G$ are given by the fractional linear substitutions $x \to \frac{ax+b}{cx+d}$ with $a, b, c, d \in K$ and $ad - bc \neq 0$.

Show that when $K$ is infinite, then $G$ is infinite and the fixed field of $G$ is $K$. Find the fixed field of $H$.

Show that when $K = F_q$ and put

then $\operatorname {ord} G = q^3 - q$ and the fixed field of $G$ is $F_q(z)$. Conclude that $F_q(x)$ is a Galois extension of a given field $L$ with $F_q \subseteq L \subseteq F_q(x)$ if and only if $L \supseteq F_q(z)$. Find the fixed field of $H$.

Show that there are unique automorphisms $\sigma$ and $\tau$ of $E/\mathbb {C}$ such that $\sigma (Y) = \zeta Y$ and $\tau (Y) = Y^{-1}$, and that the subgroup $G$ of $\operatorname {Aut}(E)$ which these generate is isomorphic to $D_n$, the Dihedral group of order $2n$.

Show that $Y$ is a root of a polynomial of degree $2n$ with coefficients in $F$.

Show that $E/F$ is a Galois extension with Galois group $G$.

Find the subgroup $G$ (of $\operatorname {Aut}_{\mathbb {Q}}(F)$) generated by $\sigma$ and $\tau$.

If $K$ is the fixed field of $G$ in $F$, find a finite subset $S$ of $F$ such that $K = \mathbb {Q}(S)$.

How many subfields lie strictly between $K$ and $F$? How many of these are Galois over $K$? Justify your answers.

Identify $G$.

Let $H$ be the subgroup of $G$ generated by $\sigma^2$ and $\tau$. Find $\alpha \in \mathbb {Q}(t)$ such that $\mathbb {Q}(\alpha )$ is the fixed field of $H$ (justify your answer).

Prove that $f(x)$ is irreducible over $\mathbb {Q}$.

Prove that $K/\mathbb {Q}$ is Galois.

Find a generator of the Galois group $\operatorname {Gal}(K/\mathbb {Q})$.

Show that the rational functions $f_a = \frac{1}{x-a}$ ($a \in K$) are linearly independent over $K$.

As $K$-modules, $K[x]$ and $K(x)$ have what dimensions?

Let $G$ denote the additive group of $K$, acting on $K(x)$ by $a \in G$ sending $x$ to $x+a$. Assume that $K$ is infinite. Let $f \in K(x)$. Show that the $G$-orbit of $f$ spans a finite-dimensional $K$-module if and only if $f \in K[x]$.

Let $K$ be a field of characteristic $p > 0$. Show that the polynomial $t^p - t - c$ in $K[t]$ is either irreducible or splits completely into $p$ linear factors over $K$.

Let $F$ be the splitting field of the polynomial $t^{62} - 1$ over $\mathbb {Z}_5$. Show that $[F : \mathbb {Z}_5] = 3$.

Suppose you are given a field $L$, $\mathbb {Q} \subseteq L \subseteq \mathbb {C}$, such that $L/\mathbb {Q}$ is algebraic and every finite field extension $K/L$, $K \subseteq \mathbb {C}$ is of even degree. Show that every finite field extension of $L$ must in fact have degree equal to a power of $2$.

Show that such a field $L$ actually exists.

Let $L$ be an intermediate field between $K^p$ and $K$. If $[L:K^p]$ is finite, prove that it is a power of $p$.

A subset $B$ of $K$ is called $p$-independent if for any finite set $b_1, b_2, \ldots , b_m$ of distinct elements of $B$

Prove that if $K^p \neq K$, then $K$ contains a maximal $p$-independent subset $B$.

Prove that the set $B$ of part (b) satisfies $K^p(B) = K$.

Prove $K$ is a field.

Let $f(x) \in K[x]$ be irreducible. Prove that $\deg (f) \geq 5$.

Show that under composition of polynomial functions, $G(k)$ forms a group.

Show that if $ax+b \in G(k)$, then the map of $k$ to itself defined by $\lambda \mapsto a\lambda +b$ is a set-theoretic automorphism of $k$.

Show that the map $G(k) \to \operatorname {aut}_{set}(k)$ defined by

is an injective group homomorphism.

Show that the subgroup $G_1(k)$ consisting of all elements of the form $x+b$ is a normal subgroup of $G(k)$, and that the map $x+b \mapsto b$ defines an isomorphism of $G_1(k)$ with the additive group of $k$.

Show that the map $ax+b \mapsto a$ defines a group homomorphism of $G(k)$ onto the multiplicative group $k^*$ whose kernel is $G_1(k)$.

Let $\Gamma \subseteq G(k)$ be a normal subgroup. Show that if $\Gamma \neq \left\{ e\right\}$, then $\Gamma$ contains $G_1(k)$.

Show that if $\Gamma \subseteq G(k)$ is a subgroup and if $\Gamma$ contains $G_1(k)$, then $\Gamma$ is a normal subgroup of $G(k)$.

Compute $\# G(F_q)$ and $\# G_1(F_q)$.

Show that $G_1(F_q)$ is the unique $p$-Sylow subgroup of $G(F_q)$.

Let $\Gamma \subseteq G(F_q)$ be a non-trivial subgroup. Show that $\Gamma$ is normal in $G(F_q)$ if and only if $q \mid \# \Gamma$.

Let $d > 1$ be a divisor of $q-1$. Show that the unique subgroup of $G(F_q)$ of order $qd$ consists of all elements $ax+b$ with $a, b$ in $F_q$ and $a^d = 1$.

Show $x^p - l$ is irreducible over $\mathbb {Q}$.

Show $K$ contains a primitive $p$-th root of unity $\zeta$.

Show $\deg (K/\mathbb {Q}) = p(p-1)$.

Fix an element $\alpha \in K$ with $\alpha^p = l$. Show that for any $\sigma \in \operatorname {Gal}(K/\mathbb {Q})$, there exist unique elements $a \in F_p^*$ and $b \in F_p$ such that

and show that the map $\sigma \mapsto ax+b$ defines an isomorphism of groups $\operatorname {Gal}(K/\mathbb {Q}) \to G(F_p)$ (see notations in 1424).

Show that the map of sets $F_p \to \left\{ \text{roots of } x^p - l \text{ in } K\right\}$ defined by $b \mapsto \zeta^b \alpha$ is bijective, and that by means of this bijection, the composite map

is the permutation action of the Galois group on the roots.

Show that if $L$ is an intermediate field, i.e., $\mathbb {Q} \subseteq L \subseteq K$ and if $L \neq K$, then $L/\mathbb {Q}$ is Galois if and only if $L \subseteq \mathbb {Q}(\zeta )$.

Show that $F_p^*$ has a unique subgroup of order $(p-1)/2$, namely the subgroup consisting of squares of elements of $F_p^*$.

Show that the map $\chi_2 : F_p^* \to \left\{ 1,-1\right\}$ (the multiplicative group) defined by

is a homomorphism (called the ``quadratic character mod $p$'').

Show that $\chi_2(a) = a^{\frac{p-1}{2}}$.

Let $\zeta_p$ be the complex number $e^{2\pi i/p}$, a primitive $p$-th root of unity in $\mathbb {C}$. So $\mathbb {Q}(\zeta_p)$ is a subfield of $\mathbb {C}$. Denote by $c : z \mapsto \overline{z}$ the automorphism ``complex conjugation'' of $\mathbb {C}$, show that $\mathbb {Q}(\zeta_p)$ is mapped to itself by $c$, and that $c$ induces an automorphism of $\mathbb {Q}(\zeta_p)$ which under the isomorphism $\operatorname {Gal}(\mathbb {Q}(\zeta_p)/\mathbb {Q}) \simeq F_p^*$ corresponds to the element $-1 \in F_p^*$.

Show that the number in $\mathbb {Q}(\zeta_p)$ defined by the ``Gauss Sum''

satisfies, as a complex number, the relations

Show that $\chi_2(-1) = 1$ if and only if $p \equiv 1 \bmod 4$.

Deduce from (e) and (f) above that

If a monic polynomial $f$ in $\mathbb {Z}[x]$ reduces modulo a prime number $p$ to an irreducible polynomial in $F_p[x]$, then $f$ is irreducible in $\mathbb {Q}^{x}$.

$x^p - x - 1$ is irreducible in $F_p[x]$.

Given any field $F$, and any finite group $G$, there exists a Galois extension $E/F$ with Galois group $G$.

If $L \supseteq E \supseteq F$ is a chain of fields with $L/E$ Galois and $E/F$ Galois, then $L/F$ is Galois.

If $F$ is a field of characteristic $p > 0$, and if $E/F$ is a finite Galois extension, then $p$ does not divide the degree of $E/F$.