What is the Galois group of a polynomial?

Definition (Galois Group): If F is the splitting field of a polynomial p(x) then G is called the Galois group of the polynomial p(x), usually written \mathrm{Gal}(p). So, taking the polynomial p(x)=x^2-2, we have G=\mathrm{Gal}(p)=\{f,g\} where f(a+b\sqrt{2})=a-b\sqrt{2} and g(x)=x.

What does Galois theory state?

The central idea of Galois’ theory is to consider permutations (or rearrangements) of the roots such that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers.

What is the order of Galois group?

The order of the Galois group equals the degree of a normal extension. Moreover, there is a 1–1 correspondence between subfields F ⊂ K ⊂ E and subgroups of H ⊂ G, the Galois group of E over F. To a subgroup H is associated the field k = {x ∈ E : f(x) = x for all f ∈ K}.

Is Galois group cyclic?

When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group.

What does Galois theory do?

In a word, Galois Theory uncovers a relationship between the structure of groups and the structure of fields. It then uses this relationship to describe how the roots of a polynomial relate to one another.

What does it mean for a group to be Galois?

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

Is Galois theory number theory?

Galois theory and algebraic number theory Galois theory is an important tool for studying the arithmetic of “number fields” (finite extensions of Q) and “function fields” (finite extensions of Fq(t)). In particular: Generalities about arithmetic of finite normal extensions of number fields and function fields.

What did Évariste Galois discover?

Évariste Galois’s most significant contribution to mathematics by far is his development of Galois theory. He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial.

What is a Galois closure?

The Galois closure of a separable field extension F/E is a minimal Galois extension over E containing F. It is unique up to isomorphism over E. When F = E(α) is a finite simple extension, its Galois closure is the splitting field of the minimal polynomial f(x) of α over E.

What is Galois extension k f?

(c) the Galois group of K/F is isomorphic to the quotient group G/H. Moreover, whether or not K/F is normal, (d) [K : F]=[G : H] and [E : K] = |H|. (3) If the intermediate field K corresponds to the subgroup H and σ is any automorphism. in G, then the field σK = {σ(x): x ∈ K} corresponds to the conjugate subgroup.

Is Galois theory important?

Galois theory is an important tool for studying the arithmetic of “number fields” (finite extensions of Q) and “function fields” (finite extensions of Fq(t)). In particular: Generalities about arithmetic of finite normal extensions of number fields and function fields.