Editing Galois group (section)

=== Finite abelian groups ===
The Galois group <math>\operatorname{Gal}(\Complex/\R)</math> has two elements, the identity automorphism and the [[complex conjugation]] automorphism.<ref>{{citation|title=Classical Algebra: Its Nature, Origins, and Uses|first=Roger L.|last=Cooke|publisher=John Wiley & Sons| year=2008| isbn=9780470277973|page=138|url=https://books.google.com/books?id=JG-skeT1eWAC&pg=PA138}}.</ref>

==== Quadratic extensions ====
The degree two field extension <math>\Q(\sqrt{2})/\Q</math> has the Galois group <math>\operatorname{Gal}(\Q(\sqrt{2})/\Q)</math> with two elements, the identity automorphism and the automorphism <math>\sigma</math> which exchanges <math>\sqrt2</math> and <math>-\sqrt2</math>. This example generalizes for a prime number <math>p \in \N.</math>

==== Product of quadratic extensions ====
Using the lattice structure of Galois groups, for non-equal prime numbers <math>p_1, \ldots, p_k</math> the Galois group of <math>\Q \left (\sqrt{p_1},\ldots, \sqrt{p_k} \right)/\Q</math> is

:<math>\operatorname{Gal} \left (\Q(\sqrt{p_1},\ldots, \sqrt{p_k})/\Q \right ) \cong \operatorname{Gal}\left (\Q(\sqrt{p_1})/\Q \right )\times \cdots \times \operatorname{Gal} \left (\Q(\sqrt{p_k})/\Q \right ) \cong (\Z/2\Z)^k</math>

==== Cyclotomic extensions ====
Another useful class of examples comes from the splitting fields of [[cyclotomic polynomial]]s. These are polynomials <math>\Phi_n</math> defined as

:<math>\Phi_n(x) = \prod_{\begin{matrix} 1 \leq k \leq n \\ \gcd(k,n) = 1\end{matrix}} \left(x-e^{\frac{2ik\pi}{n}} \right)</math>

whose degree is <math>\phi(n)</math>, [[Euler's totient function]] at <math>n</math>. Then, the splitting field over <math>\Q</math> is <math>\Q(\zeta_n)</math> and has automorphisms <math>\sigma_a</math> sending <math>\zeta_n \mapsto \zeta_n^a</math> for <math>1 \leq a < n</math> relatively prime to <math>n</math>. Since the degree of the field is equal to the degree of the polynomial, these automorphisms generate the Galois group.<ref>{{Cite book| last1=Dummit| title=Abstract Algebra| last2=Foote |pages=596, 14.5 Cyclotomic Extensions}}</ref> If <math>n = p_1^{a_1}\cdots p_k^{a_k},</math> then

:<math>\operatorname{Gal}(\Q(\zeta_n)/\Q) \cong \prod_{a_i} \operatorname{Gal}\left (\Q(\zeta_{p_i^{a_i}})/\Q \right )</math>

If <math>n</math> is a prime <math>p </math>, then a corollary of this is

:<math>\operatorname{Gal}(\Q(\zeta_p)/\Q) \cong \Z/(p-1)\Z</math>

In fact, any finite abelian group can be found as the Galois group of some subfield of a cyclotomic field extension by the [[Kronecker–Weber theorem]].

==== Finite fields ====
Another useful class of examples of Galois groups with finite abelian groups comes from finite fields. If {{math|''q''}} is a prime power, and if <math>F = \mathbb{F}_q</math> and <math>E=\mathbb{F}_{q^n}</math> denote the [[Finite field|Galois fields]] of order <math>q</math> and <math>q^n</math> respectively, then <math>\operatorname{Gal}(E/F)</math> is cyclic of order {{math|''n''}} and generated by the [[Frobenius homomorphism]].

==== Degree 4 examples ====
The field extension <math>\Q(\sqrt{2},\sqrt{3})/\Q</math> is an example of a degree <math>4</math> field extension.<ref>Since <math>\Q(\sqrt{2},\sqrt{3}) = \Q\oplus \Q\cdot\sqrt{2} \oplus \Q\cdot\sqrt{3} \oplus \Q\cdot \sqrt{6}</math> as a <math>\Q</math> vector space.</ref> This has two automorphisms <math>\sigma, \tau</math> where <math>\sigma(\sqrt{2}) = -\sqrt{2}</math> and <math>\tau(\sqrt{3})=-\sqrt{3}.</math> Since these two generators define a group of order <math>4</math>, the [[Klein four-group]], they determine the entire Galois group.<ref name=":0" />

Another example is given from the splitting field <math>E/\Q</math> of the polynomial

:<math>f(x) = x^4 + x^3 + x^2 + x + 1</math>

Note because <math>(x-1)f(x)= x^5-1,</math> the roots of <math>f(x)</math> are <math>\exp \left (\tfrac{2k\pi i}{5} \right).</math> There are automorphisms

:<math>\begin{cases}\sigma_l : E \to E \\ \sigma_2 : \exp \left (\frac{2\pi i}{5} \right) \mapsto \left (\exp \left (\frac{2\pi i}{5} \right ) \right )^l \end{cases}</math>

generating a group of order <math>4</math>. Since <math>\sigma_2</math> generates this group, the Galois group is isomorphic to <math>\Z/4\Z</math>.