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Arnav's Notes

Galois Theory - Concise Notes

MATH60037

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What is Galois Theory?

Field extensions

Definition 1. *A field homomorphism a function $\phi : K_1 \to K_2$ that preserves the field operations $\forall a,b \in K_1$

\[\phi (a+b) = \phi (a) + \phi (b), \quad \phi (ab) = \phi (a)\phi (b), \quad \phi (0_{K_1}) = 0_{K_2} \quad \phi (1_{K_1}) = \phi 1_{K_2}\]

Definition 2. $\alpha$ algebraic over $k$ if $f(\alpha ) = 0$ for some $0 \neq f \in k[X]$, otherwise $\alpha$ transcendental over $k$

Extension $k \subset K$ algebraic if $\forall \alpha \in K, \alpha$ is algebraic over $k$

Definition 3. Consider field $k$ and $f \in k[X]$. Say $k \subset K$ a splitting field for $f$ if \(f(X) = a\prod\limits_{i=1}^{n} (X- \lambda_{i}) \in K[X], \quad K = k(\lambda_1, \ldots ,\lambda_{n} )\)

Galois correspondence

Theorem 4. (Fundamental theorem of Galois Theory, Galois correspondency)

Assume characteristic 0. Let $k \subset K$ be the splitting field of $f(X) \in k[X]$ Let \(G = \{\sigma :K \to K \mid \sigma \text{ a field automorphism, } \sigma\mid_{k} = id_{k} \}\) Call this the Galois group. There is a one-to-one correspondence \(\begin{aligned} \{k \subset K_1 \subset K \mid K_1 \text{ a subfield } \} &\leftrightarrow \{H \leq G \mid H \text{ a subgroup} \}\\ K_1 &\leftrightarrow \{\sigma \in G \mid \forall k \in K_1, \sigma (\lambda ) = \lambda \\ \{\lambda \in K \mid \forall \sigma \in H, \sigma (\lambda ) = \lambda \} &\leftrightarrow \{H \leq G\} \end{aligned}\)

Definition 5. *$K \subset L$ is finite if $L$ a finite-dimensional $K$-vector space. The degree of $L$ over $K$ is \([L:K] = dim_{K}L\)

Theorem 6. (Tower Law)

Let $K \subset L \subset F$ Then

\[[F:K] = [F:L] [L:K]\]

Theorem 7. Suppose $f(X) in K[X]$ irreducible such that $f(\lambda ) =0,$ then $[K(\lambda ):K] = \mathop{deg}f$

Fundamental theorem of Galois Theory

Elementary facts

Definition 8. $K \subset L, a\in L$ . We say the evaluation homomorphism

\[e_{a}\colon K[X] \to K[a] \subset L, f(X) \mapsto f(a)\]

is a surjective ring homomorphism, where $K[a]$ the smallest subring of $L$ containing $K$ and $a$

Definition 9. $f(X) = a_0 X^n + \ldots + a_{n} \in K[X]$ is monic if $a_0 = 1$

Lemma 10. .

  • If $a$ transcendental, $e_{a}$ is injective and it extends to $\widetilde{e}_{a}:K(X) \to K(a)$ by \(DIAGRAM HERE\)

  • If $a$ algebraic then $\mathop{ker} e_{a} = \left< f_{a} \right>$ where $f_{a}\in K[X]$ irreducible or prime, and unique if $f$ monic, then called the *minimal polynomial of $a \in L / K$ . In this case \(DIARGRAM\ HERE\)

Corollary 11. For $K \subset L$ and $a \in L$ *algebraic over $K$

  • $[K(a) :K] = \mathop{deg} f_{a}$, and

  • If $K \subset F$ an extension \(\mathop{Em}_{K} (K(a), F) = \{ b \in F \mid f_{a}(b) = 0\}\)

Corollary 12. Let $K$ a field and $f \in K[X]$. Then $\exists K \subset L$ s.t $f$ has a root in $L$

REMARK TO ADD HERE

Axiomatics

Proposition 13. Fix $k \subset K$ and $k \subset L$ Then \(\# \mathop{Em}_{k} (K, L) \leq [K:k]\)

Proposition 14. Suppose given 2 field extensions $k \subset K$ and $k \subset L$. Then there is a non-unique bigger common field containing both. \(DIAGRAM\ HERE\) Formally: given $\sigma_1 \in Em(k,K)$ and $\sigma_2 \in Em(k,L)$ then $\exists \Omega , \phi_1 \in Em(k, \Omega )$ and $\phi_2 \in Em(L, \Omega )$ such that $\phi_1 \circ \sigma_1 = \phi_2 \circ \sigma_2$

Alternatively: $\exists k \subset \Omega$ such that $Em_k (K,\Omega )$ and $Em_k (L, \Omega )$ are both non-empty

Proposition 15. Let $L$ be any field and $G$ a finite group action on $L$ as automorphism. Let

\[K = G^{\ast} = \mathop{Fix} G = L^G = \{ \lambda \in L \mid \forall \sigma \in G, \sigma (\lambda ) = \lambda \}\]

Consider $\mathop{Aut}_{K}L = K^{\dagger}$. Then the obvious inclusion $G \subset K^{\dagger} = (G^{\ast} )^{\dagger}$ is an equality, so $G$ is all of $K^{\dagger}$.
Remark*

We have to contextualise half of the Galois correspondence

\[\begin{aligned} \{F \mid k \subset F \subset \Omega \} &\leftrightarrow \{G \mid G \leq \mathop{Aut}_{k}\Omega \}\\ F &\leftrightarrow \mathop{Aut}_{k} \Omega = F^{\dagger}\\ \mathop{Fix} G = G^{\ast} &\leftrightarrow G \end{aligned}\]

Lemma 16. $K \subset L$ a finite extension of degree $[L:K] \leq # G$

Galois correspondence

Definition 17. *$k \subset K$ is normal if \(\forall k \subset \Omega , \forall \sigma_1, \sigma_2 \in Em_k (K, \Omega), \exists \sigma \in Em_k (K,K), \sigma_2 = \sigma_1 \circ \sigma\) Equivalently $k \subset K$ is normal if \(\forall k \subset \Omega , \forall \sigma_1, \sigma_2 \in Em_k (K, \Omega), \sigma_2(K) \subset \sigma_1(K)\) Remark

Will see later that $k \subset K$ is normal if and only if $\exists f(X) \in k[X]$ such that $K$ a splitting field of $f$

Lemma 18. Suppose $k \subset K$ normal. Consider $k \subset L \subset K$ Then also $L \subset K$ is normal

Definition 19. $k \subset K$ is separable if $\forall k \subset K_1 \subset K_2 \subset K$, if $K_1 \neq K_2$ then $\exists k \subset \Omega$ and embeddings $x \in Em_{k}(K_1, \Omega )$ and $y_1,y_2 \in Em_{k}(K_2,\Omega )$ such that \(DIAGRAM\ HERE\) That is $y\mid_{K_1} = x$ but $y_1 \neq y_2$
We have that embeddings separate fields. Will see that

  • in Char 0, everything is separable

  • in Char $p$ there are good ways to decide if a field is separable

Lemma 20. Suppose $k \subset K \subset L$ Then $k \subset L$ separable if and only if $k\subset K, K \subset L$ is separable

Theorem 21. (Fundamental theorem of Galois theory, Galois correspondence)
Let $k \subset K$ be normal and separable. Let $G = Em_{k}(K,K)$ then there is a one-to-one correspondence

\[\begin{aligned} \{k \subset L \subset K\} &\leftrightarrow \{H \leq G\}\\ L &\rightarrow L^{\dagger} = \{\sigma \in G \mid \forall \lambda \in L, \sigma(\lambda ) = \lambda \}\\ H^{\ast} = \{\lambda \in K \mid \forall \sigma \in H, \sigma (\lambda ) = \lambda \} &\leftarrow H\end{aligned}\]

Lemma 22. *Suppose $k \subset K$ normal. Then for all towers $k \subset F \subset K \subset \Omega$, the natural restriction

\[\rho : Em_{k}(K,\Omega ) \to Em_{k}(F, \Omega )\]

is surjective

Corollary 23. *Suppose $k \subset K$ normal. Then for all towers $k \subset F \subset K \subset \Omega$

\[Em_{k}(F,K) \to Em_{k}(F,\Omega )\]

is also surjective*

Normal and separable extensions

Normal extensions

Theorem 24. For finite $k \subset K$, the following are equivalent

  1. *$\forall f \in k[X]$ irreducible, either $f$ has no roots in $K$ or $f$ splits completely in $K$

  2. *$\exists f \in k[X]$ not necessarily irreducible such that $K$ is a splitting field of $f$

  3. $k \subset K$ is normal

Proposition 25. Let $k \subset L$ be a field extension. Then there exists a tower $k \subset L \subset K$ such that $k \subset K$ is normal

Separable polynomials

Definition 26. A polynomial $f \in k[X]$ is separable if it has $n = deg(f)$ distinct roots in any field $k \subset K$ such that $f \in K[X]$ splits completely
Remark
It is not completely obvious that this definition is independent of $K$ - use the fact that 2 splitting fields are isomorphic.

Remark 27. *Derivative

\[D \colon k[X] \to k[X], X^n \mapsto nX^{n-1}\]

Having the following properties

  • *$D$ is k-linear, that is $\forall \lambda ,\mu \in k, \forall f,g \in k[X]$

    \[D(\lambda f + \mu g) = \lambda Df + \mu Dg\]

  • *Leibnitz rule, $\forall f,g \in k[X]$ \(D fg = fDg + gDf\)

Proposition 28. *$f(X) \in k[X]$ is separable if and only if $gcd(f,Df) = 1$

Lemma 29. *Let $f,g \in k[X]$ and $c = gcd(f,g) \in k[X]$
Let $k \subset L$ an extension, then $c = gcd(f,g) \in L[X]$

Theorem 30. $f\in k[X]$ irreducible is inseparable if and only if

  • $ch(k) = p > 0$, and

  • *$\exists h\in k[X]$ such that $f(X) = h(X^p)$

Definition 31. *A field $k$ in $ch (k) = p > 0$ is perfect if $\forall a \in k$ there exists $b \in k$ such that $b^p = a$

Proposition 32. If $k$ is perfect then $f \in k[X]$ is irreducible implies that $f(X)$ is separable

Definition 33. Consider $k \subset L$. An element $a \in L$ is separable over $k$ if the minimal polynomial $f(X) \in k[X]$ of $a$ is a separable polynomial

Separable degree

Definition 34. *Let $k \subset K$. Choose $K \subset \Omega$ such that $k \subset \Omega$ is normal. Define the separable degree as

\[[K:k]_{s} = \# Em_{k}(K,\Omega )\]

Remark

$[K:k]_{s}$ does not depend on $K \subset \Omega$. Suppose $k \subset \Omega_1$ and $k \subset \Omega_2$ are normal. Then there exists a bigger field $\widetilde{\Omega}$ such that $\Omega_1 \subset \widetilde{\Omega} , \Omega_2 \subset \widetilde{\Omega}$ Then

\[Em_{k}(K,\Omega_1) = Em_{k}(K,\widetilde{\Omega} ) = Em_{k}(K,\Omega_2)\]

Remark
Restate definition of separable extension. Recall $k \subset K$ separable if for all towers $k \subset K_1 \subset K_2 \subset K$ there exists $\Omega ,y : K_1 \to \Omega , x_1,x_2: K_2 \to \Omega$ such that $x_1 \neq x_2$ and $x_1\mid_{K_1} = x_2\mid_{K_2} = y$ so $[K_2:K_1] \neq 1$. Thus $k \subset K$ separable if for all towers $k\subset K_1 \subset K_2 \subset K$ $[K_2:K_1]_{s} = 1$ implies that $K_1 = K_2$

Theorem 35. *(Tower Law)
$\forall k \subset K \subset L$ \([L:K]_{s} = [L:K]_{s}[K:k]_{s}\)

Separable extensions

Recall that for $k \subset K$, said $a \in K$ separable if minimal polynomial of $f(X) \in k[X]$ of $a$ is separable polynomial

Theorem 36. *$k \subset K$ is separable if and only if $[K:k]_{s}= [K:k]$

Corollary 37. For all towers $k \subset K \subset L$ if $k \subset K$ and $K \subset L$ are separable then $k \subset L$ is separable

Corollary 38. *$k \subset K$ is separable if and only if $\forall a \in K$, $a$ is separable over $k$

Lemma 39. *Let $k \subset L \subset K$. For $\lambda \in K$, $\lambda$ is separable over $k$ implies that $\lambda$ is separable over $L$

Examples

Biquadratic extensions

Let \(K \subset K \left( \sqrt{a \pm \sqrt{b} } \right) = L, \quad c = a^{2} -b, \quad \beta = \sqrt{b} \not\in K, \quad \alpha = \sqrt{a+ \beta } \in L, \quad \alpha^\prime = \sqrt{a - \beta } \in L\) We know that $\pm \alpha ,\pm \alpha^\prime$ are roots of

\[f(X) = X^4 - 2aX^{2} +c\]

Not assuming that $f(X)$ is irreducible. Let

\[\delta = \alpha + \alpha^\prime \quad \delta^\prime = \alpha - \alpha^\prime, \quad \gamma = \alpha \alpha^\prime = \sqrt{c}\]

Then

\[\gamma^{2} =c, \quad \delta^{2} = 2(\alpha + \gamma ), \quad \delta^{2} = 2(\alpha -\gamma ),\quad \delta \delta^\prime = 2\beta, \quad \alpha = \frac{\delta +\delta^\prime }{2}, \quad \alpha^\prime = \frac{\delta -\delta^\prime }{2}\]

and we have $\pm \delta ,\pm \delta^\prime$ are roots of

\[g(Y) = Y^4 - 4aY^{2} + 4b\]

Then $L$ is the splitting field of $g$.

Assume

  1. $ch(K) \neq 2$ and

  2. $b$ is not a square in $K$ , that is $[K(\beta ):K] =2$

Claim that the extension $K \subset L$ is separable.
It is the splitting field of $f(X)$ Need to check that $gcd(f,Df) =1$ where \(Df = 4X^{3} - 4aX = 4X(X^{2} -a)\) $f,Df$ have no common roots since $X = 0$ not a root of $f$ and $X = \pm \sqrt{a}$ not a root of f, as $b \neq 0$

Theorem 40. Assume 1 and 2 from before

  1. *Suppose $bc, c$ are not square. Then

    \[[L:K] = 8, \quad G = \mathcal{D}_8\]

    and $f(X)$ is irreducible*

  2. *Suppose $bc$ square, so $c$ not square. Then

    \[[L:K] = 4, \quad G = \mathcal{C}_4\]

    and $f(X)$ is irreducible*

  3. Suppose $c$ a square, so $bc$ is not a square. Then

    • *either $2(\alpha +\gamma )$ and $2(\alpha -\gamma )$ are both not square in $K$ then

      \[[L:K] = 4, \quad G = \mathcal{C}_2 \times \mathcal{C}_2\]

      and $f(X)$ is irreducible*

    • *or one of $2(\alpha +\gamma )$ or $2(\alpha -\gamma )$ is a square in $K$, but not the other, then

      \[[L:K] = [K(\beta ):K] = 2, \quad G = \mathcal{C}_2\]

      and $f(X)$ is reducible*

Lemma 41. *Let $B \in F$ and $A \in F$ be not square in $F$. If $B$ is square in $F(\sqrt{A} )$ then either $B$ is square in $F$ or $AB$ square in $F$

Finite fields

Theorem 42. *Fix a prime $p > 0$. Then $\forall m \in \mathbb{Z}{\geq 1} \exists$ a unique, up to non-unique isomorphism, finite field with $q = p^m$ elements. The notation is $\mathbb{F}{q}$ \(G = Gal(\mathbb{F}_{q}/\mathbb{F}_{p}) = \mathbb{Z} / m \mathbb{Z}\)

Symmetric polynomials

Consider

\(f(X) = (X-x_1) \cdots (X-x_n) = X^n - \sigma_1 X^{n-1} + \cdots \pm \sigma_n \in K(x_1, \cdots ,x_n)[X]\) where \(\sigma_1 = \sigma_1(x_1, \cdots ,x_n) = \sum_{i \leq i \leq n} x_{i}, \quad \sigma_2 = \sigma_2(x_1, \cdots ,x_n) = \sum_{i \leq i \leq j \leq n} x_{i}x_{j}, \quad \cdots\) Here $\sigma_1 \in K[x_1, \ldots , x_{n} ]$ are the elementary symmetric polynomials. Let \(\delta = \prod\limits_{\text{roots of } f} (x_{i} - x_{j}), \quad \Delta = \delta^{2} = \prod\limits_{\text{roots of } f} (x_{i} - x_{j})^{2}\)

Definition 43. *$\sigma \in K[x_1, \ldots ,x_{n} ]$ is symmetric if and only if $\forall g\in S_{n}$ \(\sigma (x_{g(1)}, \ldots ,x_{g(n)}) = \sigma (x_{1}, \ldots ,x_{n})\)

Theorem 44. *Consider a degree $n$ separable polynomial $f(X) = X^n + a_1 X^{n-1} + \cdots + a_{n-1} X + a_n \in k[x]$. Let $k \subset L$ be the splitting field of $f$. Then $G \subset \mathcal{A}_{n}$ if and only if $\Delta$ is a square in $k$

Theorem 45. *Consider an irreducible cubic polynomial $X^{3} - \sigma_1 X^{2} + \sigma_2 X -\sigma_3$ and $k \subset L$ be the splitting field then $G = \mathcal{S}_3$ iff $\Delta$ is not a square in $k$, and $G = \mathcal{A}_3 = \mathcal{C}_3$ iff $\Delta$ is square in $k$

Irreducible polynomials

Proposition 46. *Suppose $f(X) = a_0 + \ldots + a_{d}X^d \in \mathbb{Z}[X]$ has a root $\frac{p}{q} \in \mathbb{Q}$ with $gcd(p,q) = 1$ then $[p \mid a_0]$ and $q \mid a_{d}$

Lemma 47. (Gauss’ Lemma)

*Suppose $f(X) = a_0 + \ldots + a_{d}X^d \in \mathbb{Z} [X]$ for $gcd(a_0, \ldots , a_{d} ) = 1$ factorises non-trivially in $\mathbb{Q} [X]$. Then it factors non-trivially in $\mathbb{Z} [X]$

Corollary 48. *if $f(X)$ is prime in $\mathbb{F}_{p}[X]$ for some $p$, then it is prime in $\mathbb{Q} [X]$

Corollary 49. (Eisenstein)

*$f(X) = a_0 + \ldots + a_{d}X^d \in \mathbb{Z} [X]$ is irreducible in $\mathbb{Q} [X]$ if there exists $p$ prime such that $\not \mid a_{d}$ but $p \mid a_{i}$ for $i < d$ and $p^{2} \not\mid a_0$

Reduction modulo prime

Theorem 50. *Let $f(X) \in \mathbb{Z} [X]$ be monic of degree $n, \mathbb{Q} \subset K$ be the splitting field of $f$ and $G = Gal(K / \mathbb{Q} ) \subset S_{n}$. For $p$ prime, denote by $\overline{f}$ as $f$ viewed in $\mathbb{F}{p}[X]$. If there exists $p$ such that $\overline{f} \in \mathbb{F}{p}[X]$ has $n$ distinct roots ina splitting field and $\overline{f} = \prod\limits_{i=1}^{k} \overline{f}{i}(X) \in \mathbb{F}{p}[X]$ with $\overline{f}{i} \in \mathbb{F}{p}[X]$ irreducible of degree $n_{i}$ then there exists $\sigma \in G \subset \mathcal{S}_{n}$ of cycle decomposition type \((n_1) \ldots (n_{k})\)

Proposition 51. *Suppose that $r$ is prime and let $G \subset \mathcal{S}{r}$ be a subgroup. If $G$ contains an $r$-cycle and one transposition then $G = \mathcal{S}{r}$

Definition 52. The character of a monoid $P$ to $K$ is $\chi : P \to K$ such that

  • $\chi (0) = 1$, and

  • *$\chi (a + b) = \chi (a) \chi (b)$ for all $a,b \in P$

Theorem 53. *Linear independence of characters, Dedekind independence theorem
Let $K$ a field and $P$ a monoid, such as $P = \mathbb{N}$. Any set of distinct non-zero characters

\[\chi_1: P \to K, \quad \ldots \quad \chi_{n}: P \to K, \ldots\]

is linearly independent in the vector space ${f:P \to K}$

Theorem 54. Let $f(X) \in \mathbb{Z} [X]$ be degree $n$ monic, $\mathbb{Q} \subset K$ be the splitting field of $f$ , $G = Gal(K / \mathbb{Q} ) \subset \mathcal{S}_{n}$ and $\lambda_1, \ldots \lambda_{n} \in K$ be the roots of $f(X)$. Let $p$ be a prime. Denote by $\overline{f}$ the image $f$ modulo $p$. Assume $\overline{f}$ is separable. Let $\mathbb{F}_{p} \subset F$ be a splitting field for $\overline{f}$, so $\overline{f}$ has $n$ distinct roots in $F$ . Let $R \subset K$ be the subring generated by the roots of $f$, so $R = \mathbb{Z} [\lambda_1, \ldots , \lambda_{n} ]$ Then

  1. *there exists a ring homomorphism $\psi : R \to F$

  2. *if $\psi^\prime : R \to F$ a ring homomorphism then $\psi^\prime$ induces a bijection \(\phi^\prime : \{ \text{roots of } f(X) \text{ in } R\} \to \{ \text{roots of } \overline{f} \text{ in } F\}\)

  3. *$\psi^\prime : R \to F$ a ring homomorphism if and only if there exists $\sigma \in G$ such that $\psi^\prime = \psi \circ \sigma$