$\DeclareMathOperator{\Q}{Q} \DeclareMathOperator{\Aut}{Aut}$

# COURSE YEAR 2017

Module - IV : July 02, 2017 – July 14, 2017, René Schoof and Laura Geatti
Topics: Computing Galois Groups, When is Gf contained in An? When is Gf transitive? Polynomials of degree at most three, Quartic poly-nomials, Examples of polynomials with Sp as Galois group over Q, Finite fields, Computing Galois groups over Q
[Homeworks]

• Construction of finite fields:
• For any prime $p$ one has $\mathbf{F}_p[x]/(f)$, with $f$ irreducible in $\mathbf{F}_p[x]$.
• Proposition $1$.
1. the cardinality of a finite field $\mathbf{K}$ of characteristic $p$ is $q=p^n$.
2. $\mathbf{K}$ contains $\mathbf{F}_p$
3. the additive group $(\mathbf{K},+)$ is isomorphic to $(\mathbf{Z}/p\mathbf{Z}$ $\times \ldots \times$ $\mathbf{Z}/p\mathbf{Z},+)$
4. the multiplicative group $(\mathbf{K}^*,\cdot )$ of a finite field is cyclic.
• Theorem $2$. For every prime power $q=p^n$ there exists a field with $q$ elements. It is of the form $\mathbf{F}[x]/(f)$ with $f$ an irreducible polynomial in $\mathbf{F}_p[X]$ of degree $n$.
• Theorem $3$. Every finite field of $q$ elements is a splitting field of $x^q-x$ over $\mathbf{F}_p$. Therefore all finite fields with $q$ elements are isomorphic. Notation $\mathbf{F}_q$.
• Examples. $\mathbf{F}_4$, $\mathbf{F}_5[\sqrt 2]$, $\mathbf{F}_9=$ $\mathbf{F}_3[i]=$ $\mathbf{F}_3[x]/(x^2+1)=$ $\mathbf{F}_3[i+1]$
• Theorem $4$. $Aut(\mathbf{F}_q) =\langle \phi\rangle$, where $\phi$ denotes the Frobenius automorphism $\phi(x)=x^p$.

• Subfields.
• Lemma $1$. Let $p$ be a prime. Let $f\in\mathbf{F}_p[x]$ be an irreducible polynomial of degree $n$. Let $\alpha$ be a zero of $f$. Then
1. $\alpha^{p^n}=\alpha$;
2. $n$ is the smallest positive integer for which (a) holds.
• Proposition $2$. Let $f\in\mathbf{F}_p[x]$ be an irreducible polynomial of degree $n$. Let $\alpha$ be a zero of $f$. Then
$f(x)=$ $(x-\alpha)$ $(x-\alpha^p)$ $\ldots$ $(x-\alpha^{p^{n-1}}).$
• Corollary $3$. Every finite extension of a finite field is a Galois extension.
• Theorem $4$. $Aut(\mathbf{F}_{p^n})$ is a cyclic group isomorphic to $\mathbf{Z}/n\mathbf{Z}$, generated by the Frobenius automorphism $\phi$.
• Proposition $5$.
1. If $\mathbf{K}$ is a subfield of $\mathbf{F}_{p^n}$, then the cardinality of $\mathbf{K}$ is equal to $p^d$, for some divisor $d$ of $n$.
2. For every divisor $d$ of $n$, there exists a unique subfield of cardinality $p^d$.
• The Galois correspondence in the case of finite fields:
Fix $p$ prime and the finite field $\mathbf{F}_{p^n}$.
$Aut( \mathbf{F}_{p^n})=$ $\langle \phi\rangle=$ $Gal(\mathbf{F}_{p^n}/\mathbf{F}_p)\cong$ $\mathbf{Z}/n\mathbf{Z}$.
For every $d$ divisor of $n$, there is a unique subfield $\mathbf{F}_{p^d}\subset \mathbf{F}_{p^n}$.
For every $d$ divisor of $n$, there is a unique subgroup $G_d$ of $Gal(\mathbf{F}_{p^n}/\mathbf{F}_p)$ of index $d$, namely the subgroup generated by $\phi^d$.
One has $\mathbf{F}_{p^d} =$ $\{x\in \mathbf{F}_{p^n}$ $~|~ g(x)=x$, $\forall g\in G_d\}$.
Conversely $G_d= \{$ $g\in Gal(\mathbf{F}_{p^n}/\mathbf{F}_p)$ $~|~g(x)=x$, $~\forall x\in \mathbf{F}_{p^d} \}$.
Example: $\mathbf{F}_{3^4}$.
• Excercise $1$.
1. Find an irreducible polynomial $f$ of degree $2$ in $\mathbf{F}_3[x]$. Then $\mathbf{F}_9=\mathbf{F}_3[x]/(f)$.
2. Which elements of $\mathbf{F}_9$ are generators of its multiplicative group $\mathbf{F}_9^*$?
3. Which elements of $\mathbf{F}_9$ have square roots in $\mathbf{F}_9$?
4. Prove that the product of all elements of $\mathbf{F}_9^*$ is 2.
5. Show that the additive group of $\mathbf{F}_9$ is not cyclic.
• Excercise $2$.
• Draw the Hasse diagrams of the subfields of each $\mathbf{F}_{2^k}$ for $k=1,..,6.$.

• Proposition $1$ The group $G_f$ permutes the roots of $f$:
if $\sigma\in G_f$ and $\alpha_i \in Zeros(f)$, then $\sigma(\alpha_i)= \alpha_j$ $\in Zeros(f).$
There is a homomorphism $\Theta\colon G_f\to S_n,$ where $S_n$ is the permutation group of $n$ elements. The homomorphism $\Theta$ is injective. Hence $\#G_f$ divides $n!$.
• Proposition $f\in\mathbf{K}[x]$ separable. Then $G_f\cong H\subset S_n$, with $H$ transitive on $\{1,2,\ldots,n\}$, if and only if $f$ is irreducible over $\mathbf{K}$.
• Criterion. $f\in\mathbf{K}[x]$ separable, $char(\mathbf{K})\not=2$. Then $G_f\cong A_n$ if and only if $Disc(f)$ is a square in $\mathbf{K}$, where $Disc(f):=$ $\prod_{i < j}$ $(\alpha_i-\alpha_j)^2\in\mathbf{K}$ (it is $G_f$-invariant).
• Example $1$. $\mathbf{K}=\mathbf{Q}$, $f(x)=$ $x^4-4=$ $(x^2+2)$ $(x^2-2)$;
$\mathbf{K}_f=\mathbf{Q} (\sqrt 2,i\sqrt 2)$, $G_f\cong \mathbf{Z}/2\mathbf{Z}$$\times \mathbf{Z}/2\mathbf{Z}. • Example 2. \mathbf{K}=\mathbf{Q}, f(x)=x^4-2; \mathbf{K}_f=\mathbf{Q}({\root 4 \of 2},i{\root 4 \of 2}), G_f\cong D_4. • Example 3. (degree 2 case). f\in\mathbf{K}[x] separable of degree 2; G_f\subset S_2=$$\{id,(12)\}$.
1. $G_f=id$ $\Leftrightarrow[\mathbf{K}_f:\mathbf{K}]$ $=1$ $\Leftrightarrow \mathbf{K}_f=K$ if and only if $f$ factors in $\mathbf{K}$.
2. $G_f=S_2 \Leftrightarrow$ $[\mathbf{K}_f:\mathbf{K}]$ $=2$ if and only if $K_f$ is a quadratic extension of $\mathbf{K}$ if and only if $f$ is irreducible over $\mathbf{K}.$
• Example $4$. (degree $3$ case). $f\in\mathbf{K}[x]$ separable of degree 3; $G_f\subset S_3$ $=$ $\{$$id,(12),(13,(23),(123),(132)$$\}$;
possible subgroups (up to conjugation): id, $S_3$, $\{id,(12)\}$, $\{ id,(123),(132)\}$.
1. $G_f=id$ $\Leftrightarrow$ $[\mathbf{K}_f:\mathbf{K}]=1$ if and only if $f$ factors in $\mathbf{K}$.
2. $G_f=S_3$ $\Rightarrow f$ is irreducible.
3. $G_f$ has order $2 \Rightarrow f$ is a product of a linear and a quadratic polynomial in $\mathbf{K}[X]$.
4. $G_f$ has order $3$ and hence $G_f=A_3 \Rightarrow f$ is irreducible.
If $char(\mathbf{K})\not=2$, we can distinguish between cases (b) and (d) using the discriminant. The Galois group $G_f$ is contained in $A_3$ if and only if $Disc(f)$ is a square in $\mathbf{K}$.
• We have done few exercises from the file Excercises $1$ ."
Table of Field $\mathbf{F}_{16}:$ download from here

• $\mathbf{K}$ a field, $f\in \mathbf{K}[x]$ a separable monic polynomial, $\mathbf{K}_f=$$\mathbf{K}[\alpha_1,\ldots,\alpha_n] the splitting field of f, \{\alpha_i\}_i zeros of f, G_f:=$$Gal(\mathbf{K}_f/\mathbf{K})$$\subset S_n the Galois group of f. • Criterion. Assume char(\mathbf{K})\not=2. Then G_f\subset A_n \Leftrightarrow Disc(f) is a square in \mathbf{K}. • Consequence. f\in \mathbf{K}[x], separable monic polynomial of degree 3, irreducible over \mathbf{K}. Then G_f=A_3, if Disc(f) is a square in \mathbf{K}, and G_f=S_3, if Disc(f) is not a square in \mathbf{K}. • The degree 4 case. f\in \mathbf{K}[x], separable monic polynomial of degree 4. 1. If f is divisible by a degree 1 factor, we are back in the degree 3 case. 2. If f factors as the product of two irreducible polynomials g, h of degree 2, then either \mathbf{K}_g=\mathbf{K}_h$$=\mathbf{K}_f$ and $G_f\cong C_2$, or $\mathbf{K}_g\not=\mathbf{K}_h$ , $\mathbf{K}_f=\mathbf{K}_g\mathbf{K}_h$ and $G_f\cong C_2\times C_2$.
3. If $f$ is irreducible, then $G_f$ is isomorphic to a transitive subgroup of $S_4$. These are $S_4$, $A_4$, $D_4$, $V_4$, $C_4$.
• The group $S_4$ and its transitive subgroups in detail.
• The cubic resolvent $g$ of $f$: its zeros $\alpha$, $\beta$, $\gamma$ are fixed by $G_f\cap V_4$, hence $g\in\mathbf{K}[x]$.
• Excercises. Computation of the Galois group of the following polynomials:
1. $x^2-101$;
2. $x^3-5x^2$$+6; 3. x^3-5x$$-5$;
4. $x^3-3x$$+1; 5. x^3-1; 6. x^3-3; 7. x^3-2x^2$$+3x+5$;

• $f\in \mathbf{K}[x]$, separable monic polynomial of degree $4$
$g$ the cubic resolvent of $f$.
$f$ separable $\Rightarrow$ $g$ separable;
$disc(f)$$=$$disc(g)$.
• Proposition. $\mathbf{K}[\alpha,\beta,\gamma]$ is the fixed field of $G_f\cap V_4$.
• Classification of the Galois groups of irreducible separable monic polynomials of degree 4, by means of the cubic resolvent:
1. $g$ irred., $disc(g)$ not square, $G_g=S_3$, then $G_f=S_4$;
2. $g$ irred., $disc(g)$ square, $G_g=A_3$, then $G_f=A_4$;
3. $g=$$h_1\cdot h_2, with deg(h_i)$$=i$, $disc(g)$ not square, $G_g=C_2$, then $G_f=D_4$$\Leftrightarrow f irred. in \mathbf{K}_g; 4. g=$$h_1\cdot h_2$, with $deg(h_i)$$=i, disc(g) not square, G_g=C_2 , then G_f=C_4$$\Leftrightarrow f$ red. in $\mathbf{K}_g$;
5. $g$ completely red., $disc(g)$ not square, $G_g=id$, then $G_f=V_4$.
• Construction of an irreducible polynomial of degree $p$, with Galois group $G_f=S_p$, for every prime $p$.
• Lemma. Let $H$ be a subgroup of $S_p$. If $H$ contains a $2$-cycle and a $p$-cycle, then $H=S_p$.
• We will construct a polynomial $f\in\mathbf{Q}[x]$, irreducible of degree $p$, with $p-2$ real roots and $2$ complex conjugate roots.
• Excercises. Computation of the Galois group of the following irreducible polynomials of degree $4$:
1. $x^4$$-x$$-1$ (type $S_4$);
2. $x^4$$+2x$$+2$ (type $S_4$);
3. $x^4$$+8x$$+12$ (type $A_4$);
4. $x^4$$+36x$$+63$ (type $V_4$);
5. $x^4$$-10x^2$$+5$ (type $C_4$);
6. $x^4$$+3x$$+3$ (type $D_4$);

• Construction of an irreducible polynomial of degree $p$, with Galois group $G_f=S_p$, for every prime~$p$.
• A method to compute Galois groups $Gal(\mathbf{K}_f/\mathbf{Q}).$ By reducing $f$ modulo various primes $p$, one computes automorphisms $\sigma_p$ in the Galois group $G_f\subset S_n$ up to conjugacy in $G_f$. The cycle decomposition of $\sigma_p$ is determined by the factorization of the polynomial $f$ modulo $p$. This method can be used to show that the Galois group $G_f$ is large. To show that it is small, other method are needed (see Milne, p.55). The method was illustrated by a computer presentation using PARI/GP.
1. Let $\mathbf{K}$ be a field of $char\not=2$. If $u,v\in\mathbf{K}^*$, then $\mathbf{K}(\sqrt u)=$$\mathbf{K}(\sqrt v) if and only if u/v is a square in \mathbf{K}^*. 2. Compute the Galois group of \mathbf{Q}_f, where f(x)=$$X^4+5x+5$ (irreducible over $\mathbf{Q}$).
3. Fix $\alpha=$$\sqrt{2-\sqrt3}$.
1. Show that $\mathbf{Q}(\alpha)$ is a normal extension of $\mathbf{Q}$.
4. Let $f$ be the polynomial $x^4-2$. Compute the Galois group of $\mathbf{K}_f$, when $\mathbf{K}=\mathbf{Q}$, $\mathbf{R}$, $\mathbf{C}$, $\mathbf{Q}(\sqrt 2)$, $\mathbf{Q}(\root4\of2)$.