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Home , Field extension, Simple extension

ADVANCED -7:BASIC THEORY OF EXTENSIONS , 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 Some Basic Definitions

2 Field Extension Theorem

3 Simple Field Extension Definition

4 Algebraic and Transcedental 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 of the field K . If R is a , we define a p(x) with coefficient in R as

2 n p(x) = a0 + a1x + a2x + ...... + anx , ∀ai ∈ R

If ai = 0 for all i, then it is known as a zero polynomial. The set of all such is denoted by R[x] and known as 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 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 √ n √ o √ n √ o

Q( 2) = a + b 2 a, b ∈ Q ,Q( 2) = a + b 2 a, b ∈ Q , is an extension field of Q,Q√, also clearly√ a simple extension. The degree is 2 because {1, 2}{1, 2} 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 {u1, ..., ud } for L over K , and a basis {w1, ..., we} 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 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 ∈ 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 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 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) √ √ √ √ Q( 2 2): Q (generated by 2 2), more generally any field (i.e., a finite extension of Q) is a simple extension Q(α) for some α. √ √ √ √ 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 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 {y ∈ N | p(y)}{y ∈ N | p(y)} 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 numbers√ 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