Abstract Algebra, Lecture 14 Jan Snellman Abstract Algebra, Lecture 14 Field extensions General field 1 extensions Jan Snellman Simple extensions 1Matematiska Institutionen Zeroes of Link¨opingsUniversitet polynomials Link¨oping,fall 2019 Lecture notes availabe at course homepage http://courses.mai.liu.se/GU/TATA55/ Abstract Algebra, Lecture 14 Jan Snellman Summary General field extensions 1 General field extensions Iterated simple extensions Simple extensions Degree, dimension Zeroes of polynomials Algebraic extensions 3 Zeroes of polynomials 2 Simple extensions Zeroes and multiplicities Classification of simple Splitting field extensions Algebraic closure Abstract Algebra, Lecture 14 Jan Snellman Summary General field extensions 1 General field extensions Iterated simple extensions Simple extensions Degree, dimension Zeroes of polynomials Algebraic extensions 3 Zeroes of polynomials 2 Simple extensions Zeroes and multiplicities Classification of simple Splitting field extensions Algebraic closure Abstract Algebra, Lecture 14 Jan Snellman Summary General field extensions 1 General field extensions Iterated simple extensions Simple extensions Degree, dimension Zeroes of polynomials Algebraic extensions 3 Zeroes of polynomials 2 Simple extensions Zeroes and multiplicities Classification of simple Splitting field extensions Algebraic closure Abstract Algebra, Lecture 14 Jan Snellman Definition Suppose that E, F are fields, and that E is a subring of F . We write E ≤ F and say that E is a subfield of F , and that F is an overfield of E. General field The inclusion map i : E F is called a field extension (or equivalently, extensions the pair E ≤ F ). Degree, dimension Algebraic extensions ! Simple extensions Example Zeroes of polynomials • Any field is an overfield of its prime subfield • Q ≤ R ≤ C • C ≤ C(x) ≤ C(x)(y) • Z2[x] Z2 ≤ (x2+x+1) Abstract Algebra, Lecture 14 Jan Snellman Definition Let E ≤ F be a field extension. Then F is a vector space over E. The dimension is denoted by [F : E], and refered to as the degree of the General field extension. If this dimension is finite, then the extension is said to be finite extensions dimensional. Degree, dimension Algebraic extensions Example Simple extensions Zeroes of polynomials • [C : R] = 2, so R ≤ C is a finite dimensional extension of degree 2. • [R : Q] = , so this extension is infinite dimensional. It is a theorem (as long as you accept the axiom of choice) that any vector space has a basis.1 In the first example, we can take f1; ig, in the second, we need set-theory yoga to produce a Hamel basis. Abstract Algebra, Lecture 14 Jan Snellman Theorem (Tower thm) If K ≤ L ≤ M, then [M : K] = [M : L][L : M]. Proof General field extensions Obvious if any extension involved is infinite, so suppose [M : L] = m < , Degree, dimension Algebraic extensions [L : K] = n < . Then M has an L-basis Simple extensions 1 u1;:::; um; Zeroes of 1 polynomials and L has a K-basis v1;:::; vn: Claim: ui vj ; 1 ≤ i ≤ m; 1 ≤ j ≤ n is a K-basis for M. Abstract Algebra, Lecture 14 Jan Snellman Proof (of claim) Spanning: take w 2 M. Then General field m extensions Degree, dimension w = ci ui ; ci 2 L Algebraic extensions i=1 X Simple extensions m 0 n 1 Zeroes of = @ dij vj A ui ; dij 2 K polynomials i=1 j=1 X X = dij vj ui i;j X Abstract Algebra, Lecture 14 Proof (of claim) Jan Snellman K-linear independence: suppose that dij vj ui = 0: i;j General field X extensions Degree, dimension Then Algebraic extensions m 0 n 1 Simple extensions @ dij vj A ui = 0; i=1 j=1 Zeroes of X X polynomials so since the ui 's are L-linearly independent, all coefficients n dij vj = 0: j=1 X But the vj 's are K-linearly independent, so all dij 's are zero. Abstract Algebra, Lecture 14 Jan Snellman Example General field [x] extensions Z2 [y] Z2[x] (x2+x+1) Degree, dimension Z2 ≤ ≤ Algebraic extensions (x2 + x + 1) (y 3 + y + 1) Simple extensions has degree 3 ∗ 2 = 6, and a basis consists of Zeroes of polynomials 1; x; y; xy; y 2; xy 2: This finite field thus has 26 = 64 elements. Abstract Algebra, Lecture 14 Jan Snellman Definition If E ≤ F , u 2 F is algebraic over E if there is a non-zero polynomial f (x) 2 E[x] having u has a zero, i.e., General field extensions n Degree, dimension i Algebraic extensions f (x) = ai x ; ai 2 E; i=0 Simple extensions X Zeroes of and polynomials n i f (u) = ai u = 0 2 F : i=0 X The smallest degree of a polynomial that works is the degree of u over E. If u is not algebraic over E, then it is transcendental over E. Abstract Algebra, Lecture 14 Jan Snellman Example p s = 2 + 1 2 R is algebraic of degree 2 over Q, since it satisfies General field 2 extensions (s - 1) - 2 = 0; Degree, dimension Algebraic extensions but no non-trivial algebraic relation of lower degree. Simple extensions On the other hand, the number Zeroes of polynomials 10-j! 1 j=1 X is transcendental over Q, as proved by Liouville. Abstract Algebra, Lecture 14 Definition Jan Snellman The extension E ≤ F is algebraic if every u 2 F is algebraic over E. Example p p Let E = , and let F = a + b 2 a; b 2 . Put u = a + b 2. General field Q Q extensions • F is a field, since Degree, dimension p Algebraic extensions p -1 1 a - b 2 a -b Simple extensions u = p = = + 2: a + b 2 a2 + 2b2 a2 + 2b2 a2 + 2b2 Zeroes of polynomials 2 2 Note that a + 2b 6= 0 when (0; 0) 6= (a; b) 2 Q × Q. • E ≤ F • E ≤ F is algebraic, with every element of F algebraic over Q with degree at most 2, since (u - a)2 - 2b2 = 0: Abstract Algebra, Lecture 14 Jan Snellman Theorem If [F : E] = n < then E ≤ F is algebraic. General field Proof. 1 extensions Degree, dimension Take u 2 F , and consider Algebraic extensions 2 n Simple extensions 1; u; u ;:::; u 2 F Zeroes of polynomials These n + 1 vectors must be linearly dependent over E, which means that there are ci 2 E, not all zero, such that n c01 + c1u + ··· + cnu = 0 Thus, u is algebraic over E. Abstract Algebra, Lecture 14 Example Jan Snellman There are algebraic extensions that are not finite-dimensional. For instance, let E = , and let F be the smallest subfield of that contains p Q R all ppfor all primesp p. Then all elements of F are algebraic; for instance, General field if u = 2 + 7 3 then extensions 12 Degree, dimension p 3 ∗ 49 49 Algebraic extensions (u - 2)2 = = 122 48 Simple extensions p 2 49 Zeroes of u - 2 2u + 2 - = 0 polynomials 48 p 49 47 2 2u = u2 + (2 - ) = u2 + 48 48 47 8u = (u2 + )2 48 p p p But the set 2; 3; 5;::: is infinite and Q-linearly independent, so [F : E] = . 1 Abstract Algebra, Lecture 14 Jan Snellman Definition Let E ≤ F be a field extension, and let u 2 F . We denote by E(u) the smallest subfield of F containing E and u, in other words General field \ extensions E(u) = K E≤K≤F Simple extensions u2K Classification of simple extensions Iterated simple Picture! extensions We can also describe it as Zeroes of polynomials p(u) E(u) = p(x); q(x) 2 E[x]; q(x) 6= z:p q(u) We call E(u) a simple extension, and u a primitive element of the extension. Abstract Algebra, Lecture 14 Jan Snellman Example p Q( 2) consists of all rational expressions like p p 2 p n a0 + a1 2 + a2 2 + ··· + an 2 General field p p 2 p m ; extensions b0 + b1 2 + b2 2 + ··· + bm 2 Simple extensions but this actually simplifies to just all Classification of simple extensions p Iterated simple extensions a0 + a1 2: Zeroes of -j! polynomials On the other hand, put u = j=1 10 , then all expressions 1 P 2 n a0 + a1u + a2u + ··· + anu 2 m ; b0 + b1u + b2u + ··· + bmu that are not identical, are different. So Q ≤ Q(u) is infinite dimensional. Abstract Algebra, Lecture 14 Theorem Jan Snellman Let E ≤ F be a field extension, and let u 2 F . 1 If u is algebraic over E, of degree n, then E ≤ E(u) is algebraic, and E[x] General field E(u) ' ; extensions p(x) Simple extensions where the minimal polynomial p(x) Classification of simple extensions 1 is irreducible Iterated simple extensions 2 has degree n Zeroes of 3 is the unique (up to association) non-zero polynomial of smallest polynomials degree such that p(u) = 0 2 If u is transcendental over E, then E ≤ E(u) is transcendental, and infinite dimensional, and E(u) ' E(x); the field of rational functions with coefficients in E. Abstract Algebra, Lecture 14 Proof Jan Snellman • Consider φ : E[x] F General field φ(f (x)) = f (u) extensions ! Simple extensions • The image is a subring of F , and is contained in E(u). In fact, it is Classification of simple extensions E[u], the smallest subring containing u. Iterated simple extensions • Let I = ker φ. Zeroes of • • If I 6= (0), then I = (p(x) for a polynomial, which is (up to polynomials association) the unique polynomial of smallest degree in I . • Of course p(u) = 0; every pol in I has u as a zero, by definition. • By the first iso thm, E[x]=I ' E[u] ⊆ E(u) ⊆ F .