Section 29 – Introduction to extension fields Instructor: Yifan Yang Spring 2007 Instructor: Yifan Yang Section 29 – Introduction to extension fields
Definition of an extension field Definition A field E is an extension field of a field if F ≤ E . Example We have F ( x , y ) C � ❅ � ❅ � ❅ � ❅ F ( x ) F ( y ) R ❅ � ❅ � ❅ � ❅ � Q F Instructor: Yifan Yang Section 29 – Introduction to extension fields
Definition of an extension field Definition A field E is an extension field of a field if F ≤ E . Example We have F ( x , y ) C � ❅ � ❅ � ❅ � ❅ F ( x ) F ( y ) R ❅ � ❅ � ❅ � ❅ � Q F Instructor: Yifan Yang Section 29 – Introduction to extension fields
Every polynomial has a zero in some extension field Remark Theorem 27.19 can be rephrased as follows. If a field is of characteristic p , then it can be regarded as an extension field of Z p . If a field is of characteristic 0, then it can be regarded as an extension field of Q . Theorem (29.3, Kronecker) Let F be a field, and f ( x ) be a nonconstant polynomial in F [ x ] . Then there exists an extension field E of F and an α ∈ E such that f ( α ) = 0 . Instructor: Yifan Yang Section 29 – Introduction to extension fields
Every polynomial has a zero in some extension field Remark Theorem 27.19 can be rephrased as follows. If a field is of characteristic p , then it can be regarded as an extension field of Z p . If a field is of characteristic 0, then it can be regarded as an extension field of Q . Theorem (29.3, Kronecker) Let F be a field, and f ( x ) be a nonconstant polynomial in F [ x ] . Then there exists an extension field E of F and an α ∈ E such that f ( α ) = 0 . Instructor: Yifan Yang Section 29 – Introduction to extension fields
Proof of Theorem 29.3 Proof. • Let p ( x ) be an irreducible factor of f ( x ) over F , say f ( x ) = p ( x ) g ( x ) for some g ( x ) ∈ F [ x ] . • By Theorems 27.9 and 27.25, F [ x ] / � p ( x ) � is a field. • The field F is naturally embedded in F [ x ] / � p ( x ) � by a �→ a + � p ( x ) � for a ∈ F . Thus, we may consider F [ x ] / � p ( x ) � as an extension field of F . • Now let α = x + � p ( x ) � ∈ F [ x ] / � p ( x ) � . • Then we have f ( α ) = f ( x ) + � p ( x ) � = p ( x ) g ( x ) + � p ( x ) � = � p ( x ) � . That is, α is a zero of f ( x ) in F [ x ] / � f ( x ) � . � Instructor: Yifan Yang Section 29 – Introduction to extension fields
Proof of Theorem 29.3 Proof. • Let p ( x ) be an irreducible factor of f ( x ) over F , say f ( x ) = p ( x ) g ( x ) for some g ( x ) ∈ F [ x ] . • By Theorems 27.9 and 27.25, F [ x ] / � p ( x ) � is a field. • The field F is naturally embedded in F [ x ] / � p ( x ) � by a �→ a + � p ( x ) � for a ∈ F . Thus, we may consider F [ x ] / � p ( x ) � as an extension field of F . • Now let α = x + � p ( x ) � ∈ F [ x ] / � p ( x ) � . • Then we have f ( α ) = f ( x ) + � p ( x ) � = p ( x ) g ( x ) + � p ( x ) � = � p ( x ) � . That is, α is a zero of f ( x ) in F [ x ] / � f ( x ) � . � Instructor: Yifan Yang Section 29 – Introduction to extension fields
Proof of Theorem 29.3 Proof. • Let p ( x ) be an irreducible factor of f ( x ) over F , say f ( x ) = p ( x ) g ( x ) for some g ( x ) ∈ F [ x ] . • By Theorems 27.9 and 27.25, F [ x ] / � p ( x ) � is a field. • The field F is naturally embedded in F [ x ] / � p ( x ) � by a �→ a + � p ( x ) � for a ∈ F . Thus, we may consider F [ x ] / � p ( x ) � as an extension field of F . • Now let α = x + � p ( x ) � ∈ F [ x ] / � p ( x ) � . • Then we have f ( α ) = f ( x ) + � p ( x ) � = p ( x ) g ( x ) + � p ( x ) � = � p ( x ) � . That is, α is a zero of f ( x ) in F [ x ] / � f ( x ) � . � Instructor: Yifan Yang Section 29 – Introduction to extension fields
Proof of Theorem 29.3 Proof. • Let p ( x ) be an irreducible factor of f ( x ) over F , say f ( x ) = p ( x ) g ( x ) for some g ( x ) ∈ F [ x ] . • By Theorems 27.9 and 27.25, F [ x ] / � p ( x ) � is a field. • The field F is naturally embedded in F [ x ] / � p ( x ) � by a �→ a + � p ( x ) � for a ∈ F . Thus, we may consider F [ x ] / � p ( x ) � as an extension field of F . • Now let α = x + � p ( x ) � ∈ F [ x ] / � p ( x ) � . • Then we have f ( α ) = f ( x ) + � p ( x ) � = p ( x ) g ( x ) + � p ( x ) � = � p ( x ) � . That is, α is a zero of f ( x ) in F [ x ] / � f ( x ) � . � Instructor: Yifan Yang Section 29 – Introduction to extension fields
Proof of Theorem 29.3 Proof. • Let p ( x ) be an irreducible factor of f ( x ) over F , say f ( x ) = p ( x ) g ( x ) for some g ( x ) ∈ F [ x ] . • By Theorems 27.9 and 27.25, F [ x ] / � p ( x ) � is a field. • The field F is naturally embedded in F [ x ] / � p ( x ) � by a �→ a + � p ( x ) � for a ∈ F . Thus, we may consider F [ x ] / � p ( x ) � as an extension field of F . • Now let α = x + � p ( x ) � ∈ F [ x ] / � p ( x ) � . • Then we have f ( α ) = f ( x ) + � p ( x ) � = p ( x ) g ( x ) + � p ( x ) � = � p ( x ) � . That is, α is a zero of f ( x ) in F [ x ] / � f ( x ) � . � Instructor: Yifan Yang Section 29 – Introduction to extension fields
Algebraic and transcendental elements Definition An element α of an extension field E of a field F is algebraic over F if f ( α ) = 0 for some nonzero polynomial f ( x ) ∈ F [ x ] . If such a polynomial does not exist, then α is transcendental over F . Example √ √ 3 • 2, i , 3 are algebraic over Q since they are zeros of x 2 − 2, x 2 + 1, and x 3 − 3, respectively. √ � • α = 1 + 2 is algebraic over Q since it satisfies ( α 2 − 1 ) 2 = 2. • π and e are transcendental over Q , although the proof is not easy. • π is algebraic over R , as it is a zero of x − π ∈ R [ x ] . Instructor: Yifan Yang Section 29 – Introduction to extension fields
Algebraic and transcendental elements Definition An element α of an extension field E of a field F is algebraic over F if f ( α ) = 0 for some nonzero polynomial f ( x ) ∈ F [ x ] . If such a polynomial does not exist, then α is transcendental over F . Example √ √ 3 • 2, i , 3 are algebraic over Q since they are zeros of x 2 − 2, x 2 + 1, and x 3 − 3, respectively. √ � • α = 1 + 2 is algebraic over Q since it satisfies ( α 2 − 1 ) 2 = 2. • π and e are transcendental over Q , although the proof is not easy. • π is algebraic over R , as it is a zero of x − π ∈ R [ x ] . Instructor: Yifan Yang Section 29 – Introduction to extension fields
Algebraic and transcendental elements Definition An element α of an extension field E of a field F is algebraic over F if f ( α ) = 0 for some nonzero polynomial f ( x ) ∈ F [ x ] . If such a polynomial does not exist, then α is transcendental over F . Example √ √ 3 • 2, i , 3 are algebraic over Q since they are zeros of x 2 − 2, x 2 + 1, and x 3 − 3, respectively. √ � • α = 1 + 2 is algebraic over Q since it satisfies ( α 2 − 1 ) 2 = 2. • π and e are transcendental over Q , although the proof is not easy. • π is algebraic over R , as it is a zero of x − π ∈ R [ x ] . Instructor: Yifan Yang Section 29 – Introduction to extension fields
Algebraic and transcendental elements Definition An element α of an extension field E of a field F is algebraic over F if f ( α ) = 0 for some nonzero polynomial f ( x ) ∈ F [ x ] . If such a polynomial does not exist, then α is transcendental over F . Example √ √ 3 • 2, i , 3 are algebraic over Q since they are zeros of x 2 − 2, x 2 + 1, and x 3 − 3, respectively. √ � • α = 1 + 2 is algebraic over Q since it satisfies ( α 2 − 1 ) 2 = 2. • π and e are transcendental over Q , although the proof is not easy. • π is algebraic over R , as it is a zero of x − π ∈ R [ x ] . Instructor: Yifan Yang Section 29 – Introduction to extension fields
Algebraic and transcendental elements Definition An element α of an extension field E of a field F is algebraic over F if f ( α ) = 0 for some nonzero polynomial f ( x ) ∈ F [ x ] . If such a polynomial does not exist, then α is transcendental over F . Example √ √ 3 • 2, i , 3 are algebraic over Q since they are zeros of x 2 − 2, x 2 + 1, and x 3 − 3, respectively. √ � • α = 1 + 2 is algebraic over Q since it satisfies ( α 2 − 1 ) 2 = 2. • π and e are transcendental over Q , although the proof is not easy. • π is algebraic over R , as it is a zero of x − π ∈ R [ x ] . Instructor: Yifan Yang Section 29 – Introduction to extension fields
Algebraic and transcendental numbers Remark The last example shows that algebraicity and transcendence depend on the ground field. So whenever we talk about algebraicity and transcendence, we should specify which field we are talking about. Definition (29.11) An element α ∈ C is an algebraic number if α is algebraic over Q . A transcendental number is an element of C that is transendental over Q . Example √ √ � 1, 2, 1 + 2 are algebraic numbers, while π and e are transcendental numbers. Instructor: Yifan Yang Section 29 – Introduction to extension fields
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