ADVANCED ALGEBRA UNIT-7:BASIC THEORY OF FIELD EXTENSIONS SIMPLE EXTENSION, ALGEBRAIC AND TRANSCEDENTAL EXTENSIONS Dr. SHIVANGI UPADHYAY ACADEMIC CONSULTANT UTTRAKHAND OPEN UNIVERSITY HALDWANI(UTTRAKHAND) [email protected] Dr. Shivangi Upadhyay Advanced Algebra 1 / 19 Overview 1 Basic theory of Field Extension Some Basic Definitions 2 Field Extension Theorem 3 Simple Field Extension Definition 4 Algebraic and Transcedental Extension Algebraic Extension Transcedental Extension Example Theorem Dr. Shivangi Upadhyay Advanced Algebra 2 / 19 Basic theory of Field Extension Field Extension Definitions A Field F is said to be embedded in a field K if F is isomorphic onto a subset of the field K . If R is a ring, we define a polynomial p(x) with coefficient in R as 2 n p(x) = a0 + a1x + a2x + ::::::: + anx ; 8ai 2 R If ai = 0 for all i, then it is known as a zero polynomial. The set of all such polynomials is denoted by R[x] and known as polynomial ring over R. Dr. Shivangi Upadhyay Advanced Algebra 3 / 19 Field Extension Field Extension Definition Let F be a field then a field K is called an extension field of F if K contains F as a subfield. Dr. Shivangi Upadhyay Advanced Algebra 4 / 19 Field Extension Dr. Shivangi Upadhyay Advanced Algebra 5 / 19 Field Extension Example The field of complex numbers CC is an extension field of the field of real numbers R;R; and RR in turn is an extension field of the field of rational numbers Q:Q: Clearly then, C=QC=Q is also a field extension. The field p n p o p n p o Q( 2) = a + b 2 a; b 2 Q ;Q( 2) = a + b 2 a; b 2 Q ; is an extension field of Q;Qp; also clearlyp a simple extension. The degree is 2 because f1; 2gf1; 2g can serve as a basis. Dr. Shivangi Upadhyay Advanced Algebra 5 / 19 Field Extension Theorem Degree of Extension Given three fields arranged in a tower, say K a subfield of L which is in turn a subfield of M, then there is a simple relation between the degrees of the three extensions L=K ; M=L and M=K : [M : K ] = [M : L] · [L : K ]:[M : K ] = [M : L] · [L : K ]: Proof. Suppose that K ; L and M form a tower of fields as in the degree formula above and that both d = [L : K ] and e = [M : L] are finite. we may select a basis fu1; :::; ud g for L over K , and a basis fw1; :::; weg for M over L. Dr. Shivangi Upadhyay Advanced Algebra 6 / 19 Field Extension Theorem We will show that the elements umwn; for m ranging through 1; 2; :::; d and n ranging through 1; 2; :::; e; form a basis for M=K Since there are precisely de of them, this proves that the dimension of M=K is de, which is the desired result. First we check that they span M=K . If x is any element of M, then since the wn form a basis for M over L, we can find elements an in L such that e X Pe x = anwn = a1w1 + ··· + aewe:x = n=1 anwn = a1w1 +···+aewe: n=1 Then, since the um form a basis for L over K , we can find elements bm, n 2 K such that for each n d X Pd an = bm;num = b1;nu1 + ··· + bd;nud :an = m=1 bm;num = m=1 b1;nu1 + ··· + bd;nud : Dr. Shivangi Upadhyay Advanced Algebra 7 / 19 Field Extension Theorem Then using the distributive law and associativity of multiplication in M we have e d ! e d X X X X x = bm;num wn = bm;n(umwn);x = n=1 m=1 n=1 m=1 Pe Pd Pe Pd n=1 m=1 bm;num wn = n=1 m=1 bm;n(umwn); which shows that x is a linear combination of the umwn with coefficients from K ; in other words they span M over K : Secondly we must check that they are linearly independent over K . So assume that e d X X Pe Pd 0 = bm;n(umwn)0 = n=1 m=1 bm;n(umwn) n=1 m=1 for some coefficients bm; n in K . Dr. Shivangi Upadhyay Advanced Algebra 8 / 19 Field Extension Theorem Using distributivity and associativity again, we can group the terms as e d ! X X Pe Pd 0 = bm;num wn;0 = n=1 m=1 bm;num wn; n=1 m=1 and we see that the terms in parentheses must be zero, because they are elements of L, and the wn are linearly independent over L. That is, d X Pd 0 = bm;num0 = m=1 bm;num m=1 for each n. Then, since the bm; n coefficients are in K , and the um are linearly independent over K . We must have that bm; n = 0 for all m and all n. This shows that the elements umwn are linearly independent over K. This concludes the proof. Dr. Shivangi Upadhyay Advanced Algebra 9 / 19 Simple Field Extension Simple Field Extension Definitions A field extension L=K is called a simple extension if there exists an element θinL with L = K (θ):L = K (θ): The element θ is called a primitive element, or generating element, for the extension; we also say that L is generated over K by θ. Dr. Shivangi Upadhyay Advanced Algebra 10 / 19 Simple Field Extension Example C : R (generated by i) p p p p Q( 2 2): Q (generated by 2 2); more generally any number field (i.e., a finite extension of Q) is a simple extension Q(α) for some α. p p p p For example, Q( 3; 7) is generated by 3 + 7: F(X): F (generated by X). Dr. Shivangi Upadhyay Advanced Algebra 11 / 19 Algebraic and Transcedental Extension Algebraic and Transcedental Extension Algebraic Extension An embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p(x) is true and the set fy 2 N j p(y)gfy 2 N j p(y)g is finite. Dr. Shivangi Upadhyay Advanced Algebra 12 / 19 Algebraic and Transcedental Extension Transcedental Extension A transcendental extension of a field k is a field extension that is not an algebraic extension. An extension K =k is transcendental if and only if the field K contains elements that are transcendental over k, that is, elements that are not roots of any non-zero polynomial with coefficients in k. Dr. Shivangi Upadhyay Advanced Algebra 13 / 19 Algebraic and Transcedental Extension Examples The field extension R=Q, that is the field of real numbers as an extension of the field of rational numbersp is transcendental while the field extensions C=R and Q( 2)=Q are algebraic, where C is the field of complex numbers. Dr. Shivangi Upadhyay Advanced Algebra 14 / 19 Algebraic and Transcedental Extension Theorem Dr. Shivangi Upadhyay Advanced Algebra 15 / 19 Algebraic and Transcedental Extension Theorem Dr. Shivangi Upadhyay Advanced Algebra 16 / 19 Algebraic and Transcedental Extension Theorem Dr. Shivangi Upadhyay Advanced Algebra 17 / 19 Algebraic and Transcedental Extension Theorem Dr. Shivangi Upadhyay Advanced Algebra 18 / 19 Algebraic and Transcedental Extension THANK YOU Dr. Shivangi Upadhyay Advanced Algebra 19 / 19.