Field Extensions and Splitting Fields Bernd Schr oder logo1 Bernd - PowerPoint PPT Presentation

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) and root extractions. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) and root extractions. The root extractions are allowed to yield elements that are not in F , but in an extension field E of F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) and root extractions. The root extractions are allowed to yield elements that are not in F , but in an extension field E of F . Note that solvability by radicals does not mean there is a general formula. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) and root extractions. The root extractions are allowed to yield elements that are not in F , but in an extension field E of F . Note that solvability by radicals does not mean there is a general formula. It means that there is some way to express the zeros of the polynomial under investigation . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family of subfields of E . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J operations in E logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity, commutativity logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity, commutativity and distributivity of multiplication � over addition hold in F j , too. j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity, commutativity and distributivity of multiplication � over addition hold in F j , too. 0 and 1 are identity elements j ∈ J for addition and multiplication in E . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity, commutativity and distributivity of multiplication � over addition hold in F j , too. 0 and 1 are identity elements j ∈ J for addition and multiplication in E . Hence they are the identity � F j . elements for addition and multiplication in j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof (concl.). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then y = y + 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . � Because j ∈ J was arbitrary, we conclude that − x ∈ F j . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . � Because j ∈ J was arbitrary, we conclude that − x ∈ F j . Thus j ∈ J � F j contains additive inverses. j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . � Because j ∈ J was arbitrary, we conclude that − x ∈ F j . Thus j ∈ J � F j contains additive inverses. Multiplicative inverses are j ∈ J handled similarly. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . � Because j ∈ J was arbitrary, we conclude that − x ∈ F j . Thus j ∈ J � F j contains additive inverses. Multiplicative inverses are j ∈ J handled similarly. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . Now let F be a field, let p ∈ F [ x ] and let E be an extension field in which p splits. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . Now let F be a field, let p ∈ F [ x ] and let E be an extension field in which p splits. Then the field � S : = { D : D is an extension field of F , D ⊆ E , p splits in D } logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . Now let F be a field, let p ∈ F [ x ] and let E be an extension field in which p splits. Then the field � S : = { D : D is an extension field of F , D ⊆ E , p splits in D } is called the splitting field for p over F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . Now let F be a field, let p ∈ F [ x ] and let E be an extension field in which p splits. Then the field � S : = { D : D is an extension field of F , D ⊆ E , p splits in D } is called the splitting field for p over F . (We will address the fact that we say “the” in the next presentation.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums, products logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums, products, additive inversions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums, products, additive inversions and multiplicative inversions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums, products, additive inversions and multiplicative inversions (except divisions by zero). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. Now r logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. Now r = r 1 + r 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. Now r = r 1 + r 2 = p 1 + p 2 q 1 q 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. Now = p 1 q 2 + p 2 q 1 r = r 1 + r 2 = p 1 + p 2 . q 1 q 2 q 1 q 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof (concl.). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. It is now easy to verify that the rational combinations are a field logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. It is now easy to verify that the rational combinations are a field and that every subfield of E that contains F ∪{ θ 1 ,..., θ n } contains all rational combinations of the θ j . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. It is now easy to verify that the rational combinations are a field and that every subfield of E that contains F ∪{ θ 1 ,..., θ n } contains all rational combinations of the θ j . (Good exercise.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. It is now easy to verify that the rational combinations are a field and that every subfield of E that contains F ∪{ θ 1 ,..., θ n } contains all rational combinations of the θ j . (Good exercise.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

Subfields Splitting Fields Adjoining Elements Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields