Normal extension  In abstract algebra, a normal extension is an algebraic field extension "L"/"K" for which every polynomial that is irreducible over "K" either has no root in "L" or splits into linear factors in "L". Bourbaki calls such an extension a quasi-Galois extension. The algebraic field extension "L"/"K" is normal (we also say that "L" is normal over "K") if every irreducible polynomial over K that has at least one root in "L" splits over "L". In other words, if "α" ∈ "L", then all conjugates of "α" over "K" (i.e., all roots of the minimal polynomial of "α" over "K") belong to "L". The normality of "L"/"K" is equivalent to either of the following properties. Let "K""a" be an algebraic closure of "K" containing "L". If "L" is a finite extension of "K" that is separable (for example, this is automatically satisfied if "K" is finite or has characteristic zero) then the following property is also equivalent:   Let "L" be an extension of a field "K". Then:   For example, formula_1 is a normal extension of formula_2 since it is a splitting field of formula_3 On the other hand, formula_4 is not a normal extension of formula_5 since the irreducible polynomial formula_6 has one root in it (namely, formula_7), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field formula_8 of algebraic numbers is the algebraic closure of formula_9 i.e., it contains formula_10 Since,   and, if "ω" is a primitive cubic root of unity, then the map  is an embedding of formula_4 in formula_8 whose restriction to formula_15 is the identity. However, σ is not an automorphism of formula_16. For any prime , the extension formula_17 is normal of degree . It is a splitting field of . Here formula_18 denotes any th primitive root of unity. The field formula_19 is the normal closure (see below) of formula_16. 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".