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. 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 Ka 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, Q ( 2 ) {displaystyle mathbb {Q} ({sqrt {2}})} is a normal extension of Q , {displaystyle mathbb {Q} ,} since it is a splitting field of x 2 − 2. {displaystyle x^{2}-2.} On the other hand, Q ( 2 3 ) {displaystyle mathbb {Q} ({sqrt{2}})} is not a normal extension of Q {displaystyle mathbb {Q} } since the irreducible polynomial x 3 − 2 {displaystyle x^{3}-2} has one root in it (namely, 2 3 {displaystyle {sqrt{2}}} ), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field Q ¯ {displaystyle {overline {mathbb {Q} }}} of algebraic numbers is the algebraic closure of Q , {displaystyle mathbb {Q} ,} i.e., it contains Q ( 2 3 ) . {displaystyle mathbb {Q} ({sqrt{2}}).} Since, and, if ω is a primitive cubic root of unity, then the map is an embedding of Q ( 2 3 ) {displaystyle mathbb {Q} ({sqrt{2}})} in Q ¯ {displaystyle {overline {mathbb {Q} }}} whose restriction to Q {displaystyle mathbb {Q} } is the identity. However, σ is not an automorphism of Q ( 2 3 ) {displaystyle mathbb {Q} ({sqrt{2}})} . For any prime p, the extension Q ( 2 p , ζ p ) {displaystyle mathbb {Q} ({sqrt{2}},zeta _{p})} is normal of degree p(p − 1). It is a splitting field of xp − 2. Here ζ p {displaystyle zeta _{p}} denotes any pth primitive root of unity. The field Q ( 2 3 , ζ 3 ) {displaystyle mathbb {Q} ({sqrt{2}},zeta _{3})} is the normal closure (see below) of Q ( 2 3 ) {displaystyle mathbb {Q} ({sqrt{2}})} .