In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, an algebraic field extension

Algebraic extension

In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...

 L/K is said to be normal if L is the splitting field

Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...

 of a family of polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s in K[X]. Bourbaki

Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...

 calls such an extension a quasi-Galois extension

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

.

Equivalent properties and examples

The normality of L/K is equivalent to each of the following properties:

• Let K^a be an algebraic closure
  Algebraic closure
  In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
  
   of K containing L. Every embedding
  Embedding
  In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
  
   σ of L in K^a which restricts to the identity on K, satisfies σ(L) = L. In other words, σ is an automorphism
  Automorphism
  In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
  
   of L over K.
• Every irreducible polynomial
  Irreducible polynomial
  In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
  
   in K[X] which has a root in L factors into linear factors in L[X].
• The minimal polynomial
  Minimal polynomial (field theory)
  In field theory, given a field extension E / F and an element α of E that is an algebraic element over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p = 0...
  
   over K of every element in L splits
  Splitting field
  In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
  
   over L.

For example, is a normal extension of , since it is a splitting field of x² − 2. On the other hand, is not a normal extension of since the polynomial x³ − 2 has one root in it (namely, ), but not all of them (it does not have the non-real cubic roots of 2).

The fact that is not a normal extension of can also be proved using the first of the two equivalent properties from above. The field of complex algebraic number

Algebraic number

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

s is an algebraic closure of containing . On the other hand
and, if ω is one of the two non-real cubic roots of 2, then the map
is an embedding of in whose restriction to is the identity. However, σ is not an automorphism of .

For any prime p, the extension is normal of degree p(p − 1). It is a splitting field of x^p − 2. Here denotes any pth primitive root of unity.

Other properties

Let L be an extension of a field K. Then:

• If L is a normal extension of K and if E is an intermediate extension (i.e., L ⊃ E ⊃ K), then L is also a normal extension of E.

• If E and F are normal extensions of K contained in L, then the compositum EF and E ∩ F are also normal extensions of K.

Normal closure

If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is a normal extension of K. Furthermore, up to isomorphism there is only one such extension which is minimal, i.e. such that the only subfield of M which contains L and which is a normal extension of K is M itself. This extension is called the normal closure of the extension L of K.

If L is a finite extension of K, then its normal closure is also a finite extension.