School of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information

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1 MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon (odd), Wed & Fri 11am, Maths Room 1A The tutorials begin in Week 2. There are two choices of tutorial time: Mon 2pm (Maths Room 1C), Fri 2pm (Maths Room 1C) Assessment: 100% exam Fire Safety: If the fire alarm were to be sounded during a lecture, the route to exit from Room 1A is to leave via the back-door of the Maths Institute. Recommended Texts: Ian Stewart, Galois Theory, Chapman & Hall; 3rd Edition, 2004 in the library; 4th Edition, John M. Howie, Fields and Galois Theory, Springer Undergraduate Mathematics Series, Springer, P. M. Cohn, Algebra, Vol. 2, Wiley, 1977, Chapter 3. [Out of print, but available in the library.] Support Materials: All handouts, problem sheets and solutions will be available in PDF format from MMS. Solutions will only be posted in MMS after all relevant tutorials have happened. The handouts should contain all information appearing on slides. 1

2 Course Content: Basic facts about fields and polynomial rings: Mostly a review of material from MT3505, but some new information about irreducible polynomials. Field extensions: Terminology and basic properties about the situation of two fields with F K. Splitting fields & normal extensions: Field extensions constructed by adjoining the roots of a polynomial, constructed so that this polynomial factorizes into linear factors over the larger field. Basic facts about finite fields: Existence and uniqueness of fields of order p n, plus the fact that the multiplicative group of a finite field is cyclic. Separable extensions & the Theorem of the Primitive Element: Separability: A technical condition to avoid repeated roots of irreducible polynomials. The Theorem of the Primitive Element applies in this circumstance and allows us to assume that our field extensions have a specific form and hence to simplify various proofs. Galois groups & the Fundamental Theorem of Galois Theory: The definition of the Galois group as the collection of invertible structure preserving maps of a field extension. The Fundamental Theorem of Galois Theory states that the structure of the Galois group corresponds to the structure of the field extension. Applications: Specifically the link between solution of the polynomial equation by radicals to solubility of the Galois group. 2

3 MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout I: Rings, Fields and Polynomials 1 Rings, Fields and Polynomials The first section of the module contains a review of the background material on rings, fields, polynomials, polynomial rings and the irreducibility of polynomials. The majority comes from the module MT3505 Rings and Fields. Rings Definition 1.1 A commutative ring with a 1 is a set R endowed with two binary operations denoted as addition and multiplication such that the following conditions hold: (i) R forms an abelian group with respect to addition (with additive identity 0, called zero); (ii) multiplication is associative: a(bc) = (ab)c for all a, b, c R; (iii) multiplication is commutative: ab = ba for all a, b R; (iv) the distributive laws hold: for all a, b, c R; a(b + c) = ab + ac (a + b)c = ac + bc (v) there is a multiplicative identity 1 in R satisfying a1 = 1a = a for all a R. Definition 1.2 Let R be a commutative ring with a 1. An ideal I in R is a non-empty subset of R that is both an additive subgroup of R and satisfies the property that if a I and r R, then ar I. Thus a subset I of R is an ideal if it satisfies the following four conditions: (i) I is non-empty (or 0 I); (ii) a + b I for all a, b I; (iii) a I for all a I; (iv) ar I for all a I and r R. 1

4 Let R be a commutative ring and let I be an ideal of R. Then I is, in particular, a subgroup of the additive group of R and the latter is an abelian group. We can therefore form the additive cosets of I; that is, define I + r = { a + r a I } for each r R. We know from group theory when two such cosets are equal, I + r = I + s if and only if r s I, and that the set of all cosets forms a group via addition of the representatives: (I + r) + (I + s) = I + (r + s) for r, s R. The assumption that I is an ideal then ensures that there is a well-defined multiplication on the set of cosets, given by (I + r)(i + s) = I + rs for r, s R, with respect to which the set of cosets I + r forms a ring, called the quotient ring and denoted by R/I. Theorem 1.3 Let R be a commutative ring with a 1 and I be an ideal of R. Then the quotient ring R/I is a commutative ring with a 1. Definition 1.4 Let R and S be commutative rings with 1. A homomorphism φ: R S is a map such that (i) (a + b)φ = aφ + bφ (ii) (ab)φ = (aφ)(bφ) for all a, b R. Definition 1.5 Let R and S be commutative rings with 1 and φ: R S be a homomorphism. (i) The kernel of φ is (ii) The image of φ is ker φ = { a R aφ = 0 }. im φ = Rφ = { aφ a R }. Theorem 1.6 (First Isomorphism Theorem) Let R and S be commutative rings with 1 and φ: R S be a homomorphism. Then the kernel of φ is an ideal of R, the image of φ is a subring of S and R/ker φ = im φ. Definition 1.7 Let R be a commutative ring with a 1. (i) A zero divisor in R is a non-zero element a such that ab = 0 for some non-zero b R. (ii) An integral domain is a commutative ring with a 1 containing no zero divisors. 2

5 Fields Definition 1.8 A field F is a commutative ring with a 1 such that 0 1 and every non-zero element is a unit, that is, has a multiplicative inverse. Proposition 1.10 (i) Every field is an integral domain. (ii) The set of non-zero elements in a field forms an abelian group under multiplication. We write F for the multiplicative group of non-zero elements in a field. If F is any field, with multiplicative identity denoted by 1, and n is a positive integer, let us define n = 1 } {{ + 1 }. n times By the distributive law, mn = m n for all positive integers m and n. Since F is, in particular, an integral domain, it follows that if there exists a positive integer n such that n = 0 then necessarily the smallest such positive integer n is a prime number. Definition 1.11 Let F be a field with multiplicative identity 1. (i) If it exists, the smallest positive integer p such that p = 0 is called the characteristic of F. (ii) If no such positive integer exists, we say that F has characteristic zero. Our observation is therefore that every field F either has characteristic zero or has characteristic p for some prime number p. We shall say that K is a subfield of F when K F and that K forms a field itself under the addition and multiplication induced from F ; that is, when the following conditions hold: (i) K is non-empty and contains non-zero elements (or, equivalently when taken with the other two conditions, 0, 1 K); (ii) a + b, a, ab K for all a, b K; (iii) 1/a K for all non-zero a K. Theorem 1.12 Let F be a field. (i) If F has characteristic zero, then F has a unique subfield isomorphic to the rationals Q and this is contained in every subfield of F. (ii) If F has characteristic p (prime), then F has a unique subfield isomorphic to the field F p of integers modulo p and this is contained in every subfield of F. Definition 1.13 This unique minimal subfield in F is called the prime subfield of F. 3

6 Polynomials Definition 1.14 Let F be a field. A polynomial over F in the indeterminate X is an expression of the form f(x) = a 0 + a 1 X + a 2 X a n X n where n is a non-negative integer and the coefficients a 0, a 1,..., a n are elements of F. We shall write F [X] for the set of all polynomials in the indeterminate X with coefficients taken from the field F. Proposition 1.15 If F is a field, the polynomial ring F [X] is a Euclidean domain. The Euclidean function associated to F [X] is the degree of a polynomial. If f(x) = a n X n + a n 1 X n 1 + +a 1 X+a 0 is a non-zero polynomial with leading term having non-zero coefficient, that is, a n 0, the degree of f(x) is The properties of the degree are: deg f(x) = n. (i) if f(x) and g(x) are non-zero, then deg f(x)g(x) = deg f(x) + deg g(x); (ii) if f(x) and g(x) are polynomials with f(x) 0, then there exist unique polynomials q(x) and r(x) satisfying g(x) = q(x) f(x) + r(x) with either r(x) = 0 or deg r(x) < deg f(x). Proposition 1.16 If F is a field, the polynomial ring F [X] is a principal ideal domain; that is, every ideal I in F [X] has the form I = (f(x)) = { f(x)g(x) g(x) F [X] } for some polynomial f(x). Definition 1.17 Suppose f(x) and g(x) are polynomials over the field F. A greatest common divisor of f(x) and g(x) is a polynomial h(x) of greatest degree such that h(x) divides both f(x) and g(x). To say that h(x) divides f(x) means that f(x) is a multiple of h(x); that is, f(x) = h(x) q(x) for some q(x) F [X]. In a general Euclidean domain, the greatest common divisor is defined uniquely up to multiplication by a unit. In the polynomial ring F [X], the units are constant polynomials (that is, elements of the base field F viewed as elements of F [X]). As a consequence, the greatest common divisor of a pair of polynomials is defined uniquely up to multiplication by a scalar from the field F. Theorem 1.18 Let F be a field and f(x) and g(x) be two non-zero polynomials over F. Then there exist u(x), v(x) F [X] such that the greatest common divisor of f(x) and g(x) is given by h(x) = u(x) f(x) + v(x) g(x). 4

7 Definition 1.19 Let f(x) be a polynomial over a field F of degree at least 1. We say that f(x) is irreducible over F if it cannot be factorized as f(x) = g 1 (X) g 2 (X) where g 1 (X) and g 2 (X) are polynomials in F [X] of degree smaller than f(x). The term reducible is used for a polynomial that is not irreducible; that is, that can be factorized as a product of two polynomials of smaller degree. Theorem 1.22 (Gauss s Lemma) Let f(x) be a polynomial with integer coefficients. Then f(x) is irreducible over Z if and only if it is irreducible over Q. Theorem 1.23 (Eisenstein s Irreducibility Criterion) Let f(x) = a 0 + a 1 X + + a n X n be a polynomial over Z. Suppose there exists a prime number p such that (i) p does not divide a n ; (ii) p divides a 0, a 1,..., a n 1 ; (iii) p 2 does not divide a 0. Then f(x) is irreducible over Q. Every integral domain has a field of fractions. To be precise, if R is an integral domain, the field of fractions of R is the set of all expressions of the form r/s where (i) r, s R with s 0, and (ii) we define r 1 /s 1 = r 2 /s 2 if and only if r 1 s 2 = r 2 s 1. We define r 1 s 1 + r 2 s 2 = r 1s 2 + r 2 s 1 s 1 s 2 and r 1 s 1 r2 s 2 = r 1r 2 s 1 s 2 for such fractions r 1 /s 1 and r 2 /s 2. With respect to these operations, the set of all fractions r/s forms a field. Finally R embeds in the field of fractions via the map r r/1; that is, the set { r/1 r R } is a subring isomorphic to the original integral domain R. We apply this construction in the case when R = F [X], the integral domain of polynomials with coefficients from the field F : Definition 1.24 Let F be a field. The field of rational functions with coefficients in F is denoted by F (X) and is the field of fractions of the polynomial ring F [X]. The elements of F (X) are expressions of the form f(x) g(x) where f(x) and g(x) are polynomials with coefficients from F. Proposition 1.25 The field F occurs as a subfield of the field F (X) of rational functions with coefficients in F. 5

8 MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout II: Field Extensions: Algebraic elements, minimum polynomials, and simple extensions 2 Field Extensions Definition 2.1 Let F and K be fields such that F is a subfield of K. We then say that K is an extension of F. We also call F the base field of the extension. In particular, note that every field is an extension of its prime subfield. The point of this definition, though, is a change of perspective. We are not viewing a field extension F K as being the situation where we start with a field K and then pass to a subfield F. Instead, the philosophy here will be much more starting with a base field F and then creating a bigger field K containing F that is the extension. The degree of an extension The first observation to make in this setting is that if the field K is an extension of the field F, then K, in particular, satisfies the following conditions: K forms an abelian group under addition; we can multiply elements of K by elements of F ; a(x + y) = ax + ay for all a F and x, y K; (a + b)x = ax + bx for all a, b F and x K; (ab)x = a(bx) for all a, b F and x K; 1x = x for all x K. Thus, we can view K as a vector space over the field F. Definition 2.2 Let the field K be an extension of the field F. (i) The degree of K over F is the dimension of K when viewed as a vector space over F. We denote this by K : F. Thus K : F = dim F K. (ii) If the degree K : F is finite, we say that K is a finite extension of F. Warning: Note that saying K is a finite extension of F does not mean that K is a finite field. There are many situations where both fields have infinitely many elements in them. It refers precisely to the dimension of the bigger field over the smaller field. Theorem 2.4 (Tower Law) Let F K L be field extensions. Then L is a finite extension of F if and only if L is a finite extension of K and K is a finite extension of F. In such a case, L : F = L : K K : F. 1

9 Comment: In the course of the proof of the theorem we observe that if F K L are field extensions, {v 1, v 2,..., v m } is a basis for L over K and {w 1, w 2,..., w n } is a basis for K over F, then { v i w j 1 i m, 1 j n } is a basis for L over F. Algebraic elements and algebraic extensions Definition 2.5 Let the field K be an extension of the field F. (i) An element α K is said to be algebraic over F if there exists a non-zero polynomial f(x) F [X] such that f(α) = 0. When this holds, we shall say that α satisfies the polynomial equation f(x) = 0. (ii) We say that K is an algebraic extension of F if every element of K is algebraic over F. Thus to say that an element α K is algebraic over the subfield F is to say that there are coefficients b 0, b 1,..., b n in F such that b 0 + b 1 α + b 2 α b n α n = 0. Lemma 2.6 Every finite extension is an algebraic extension. The example at the end of this chapter shows that the converse does not hold: there are algebraic extensions that are not finite extensions. Simple extensions Definition 2.7 Let the field K be an extension of the field F and α 1, α 2,..., α n be elements of K. We write F (α 1, α 2,..., α n ) for the smallest subfield of K that contains both F and the elements α 1, α 2,..., α n. It is straightforward to verify that the intersection of a collection of subfields of K is again a subfield. Consequently, the smallest subfield containing F and the elements α 1, α 2,..., α n makes sense: it is the intersection of all the subfields of K that contain this collection of elements. Definition 2.8 We say that the field K is a simple extension of the field F if K = F (α) for some α K. We then also say that K is obtained by adjoining the element α to F. Example 2.9 Let F be a field and X be an indeterminate. The field F (X) of rational functions is a simple extension of F. The indeterminate X is not an algebraic element over F. We use the term transcendental for an element that is not algebraic over the base field. It turns out that if α is any transcendental element over the base field F, then the simple extension F (α) is isomorphic to the field F (X) of rational functions. 2

10 Minimum polynomials Definition 2.10 Let F be a field and α be an element in some field extension of F such that α is algebraic over F. The minimum polynomial of α over F is the monic polynomial f(x) of least degree in F [X] such that f(α) = 0. Recall that a polynomial is monic if its leading term has coefficient 1. Lemma 2.11 Let F be a field and α be an element in some field extension of F such that α is algebraic over F. Then (i) the minimum polynomial of α over F exists; (ii) the map φ: F [X] F (α) given by g(x) g(α) (that is, evaluating each polynomial at α) is a ring homomorphism with kernel ker φ = ( f(x) ) ; (iii) the minimum polynomial f(x) of α over F is irreducible over F ; (iv) if g(x) F [X], then g(α) = 0 if and only if the minimum polynomial f(x) of α over F divides g(x); (v) the minimum polynomial f(x) of α over F is unique; (vi) if g(x) is any monic polynomial over F such that g(α) = 0, then g(x) is the minimum polynomial of α over F if and only if g(x) is irreducible over F. Comment: Note that the minimum polynomial of an algebraic element α depends upon the particular base field. For example, 2 has minimum polynomial over X 2 2 over Q, whereas its minimum polynomial over R is X 2. If we concentrate our efforts on simple extensions F (α) with α algebraic over the base field F, there are two questions that naturally arise and whose answers will enable us to make progress: (i) Given an irreducible polynomial f(x) over the field F, can we construct a simple extension F (α) such that the minimum polynomial of α over F is f(x)? (ii) If α is algebraic over F, what is the structure of the simple extension F (α) and in what way is this determined by the minimum polynomial of α over F? These questions essentially boil down to the existence of simple extensions and to then investigating their properties (and essentially establishing uniqueness as a consequence). Note that in answering the first question in the affirmative, as we do in the following theorem, we are showing that we can always adjoin a root α of an irreducible polynomial to a field F to construct some simple extension F (α). Theorem 2.13 Let F be a field and f(x) be a monic irreducible polynomial over F. Then there exists a simple extension F (α) of F such that α is algebraic over F with minimum polynomial f(x). 3

11 Comments: In the proof of the above theorem, we construct F (α) as the quotient ring K = F [X]/I, where I = (f(x)) is the ideal generated by the polynomial f(x). There are two comments to make placing the above existence result for simple extensions in context. (i) The Correspondence Theorem for rings tells us that there is a one-one correspondence between ideals in the quotient ring F [X]/I, where I = (f(x)), and ideals in the polynomial ring F [X] that contain I. We have shown that when f(x) is irreducible, the quotient K = F [X]/I is a field; that is, it has only two ideals 0 and K itself. Therefore, via the correspondence, I = (f(x)) is a maximal ideal of the polynomial ring: there are no ideals J satisfying I < J < F [X]. Consequently, we are observing above that (f(x)) is a maximal ideal when f(x) is irreducible. (The implication also reverses, as follows quite easily, but we omit the proof.) (ii) Recall that the prime subfields are constructed from the ring of integers Z. We observed, in Theorem 1.12, that the prime subfield of any field is either isomorphic to Q (which is the field of fractions of the Euclidean domain Z) or to a finite field F p (which occurs as the quotient Z/(p) by the ideal generated by some prime p, the primes being the irreducible elements in Z). An analogous observation is being made here. If F is a field, the simple extensions of F are constructed from the Euclidean domain F [X] as follows: If α is transcendental, then F (α) is isomorphic to the field of fractions, F (X), of F [X]. If α is algebraic, then F (α) can be constructed as the quotient F [X]/(f(X)) by an ideal generated by an irreducible polynomial f(x). Theorem 2.14 Let F be a field and α be an element in some extension of F. The simple extension F (α) over F is a finite extension if and only if α is algebraic over F. Moreover, in this case, F (α) : F = deg f(x), the degree of the minimum polynomial f(x) of α over F. Furthermore, (as rings). F (α) = F [X] (f(x)) Corollary 2.15 Suppose that α is algebraic over F with minimum polynomial of degree n. Then {1, α, α 2,..., α n 1 } is a basis for the simple extension F (α) over F. Theorem 2.17 Let K be an extension of a field F. Then K is a finite extension of F if and only if K = F (α 1, α 2,..., α n ) for some finite collection α 1, α 2,..., α n of elements of K each of which is algebraic over F. Example 2.19 Let us write A for the set of all elements of C that are algebraic over Q. We call A the field of algebraic numbers over Q. In this example, we show that A is indeed a subfield of C and determine the degree A : Q. 4

12 MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout III: Splitting Fields and Normal Extensions 3 Splitting Fields and Normal Extensions Splitting fields Let F be a field and consider a polynomial f(x) over the field F. Suppose that there is an extension L of F such that, when f(x) is viewed as a polynomial over L, we can factorize it as a product of linear factors: f(x) = c(x α 1 )(X α 2 )... (X α n ). We shall then say that f(x) splits over L. Necessarily, in such a situation, then the roots α 1, α 2,..., α n of f(x) are elements of the field L. Definition 3.1 Let f(x) be a polynomial over some field F. We say that a field K is a splitting field for f(x) over F if K is an extension of F satisfying the following properties: (i) f(x) splits into a product of linear factors over K, and (ii) if F L K and f(x) splits over L, then L = K. Thus, a splitting field for a polynomial f(x) over a field F is an extension K of F in which f(x) splits over K but such that f(x) does not split over any proper subfield of K; that is, K is a minimal field over which f(x) splits. Lemma 3.2 Let f(x) be a polynomial over a field F and suppose there is some extension L of F such that f(x) splits over L with roots α 1, α 2,..., α n. Then is a splitting field for f(x) over F. K = F (α 1, α 2,..., α n ) In particular, in the case of a polynomial f(x) over F = Q, we know that L = C is a suitable extension to use in the lemma since we know from the Fundamental Theorem of Algebra (proved in Complex Analysis) that every polynomial over Q has roots in C and hence splits over C. We then obtain a splitting field for f(x) over Q as Q(α 1, α 2,..., α n ) where α 1, α 2,..., α n are the roots of f(x) in C. 1

13 Existence of splitting fields Theorem 3.4 (Existence of Spitting Fields) Let f(x) be a polynomial of degree n over a field F. Then there is a splitting field K for f(x) over F with degree K : F dividing n!. Uniqueness of splitting fields and related isomorphisms Lemma 3.5 Let φ: F 1 F 2 be an isomorphism between two fields. Let f(x) be an irreducible polynomial in F 1 [X] and write f φ (X) for the polynomial over F 2 obtained by applying φ to the coefficients in f(x). Let α be a root of f(x) and β be a root of f φ (X) in some extensions of F 1 and F 2, respectively. Then there exists an isomorphism ψ : F 1 (α) F 2 (β) which extends φ and maps α to β. To say that ψ extends φ means that aψ = aφ for all a F 1 ; that is, the restriction ψ F1 of ψ to F 1 is the isomorphism φ we started with. Theorem 3.6 Let φ: F 1 F 2 be an isomorphism between two fields. Let f(x) be any polynomial in F 1 [X] and write f φ (X) for the polynomial over F 2 obtained by applying φ to the coefficients in f(x). Let K 1 be a splitting field for f(x) over F 1 and K 2 be a splitting field for f φ (X) over F 2. Then there exists an isomorphism θ : K 1 K 2 which extends φ. To establish uniqueness of splitting fields, we take F 1 = F 2 = F and φ to be the identity map in the above theorem. We phrase this uniqueness in terms of the following definition: Definition 3.7 Let F be a field and let K 1 and K 2 be extensions of F. An F -isomorphism from K 1 to K 2 is a field isomorphism ψ : K 1 K 2 such that We then say K 1 and K 2 are F -isomorphic. aψ = a for all a F. Corollary 3.8 (Uniqueness of Splitting Fields) Let f(x) be a polynomial over a field F. Any two splitting fields for f(x) over F are F -isomorphic. Definition 3.9 (i) An automorphism of a field F is an isomorphism from F to itself. (ii) Let K be an extension of the field F. An F -automorphism of K is an F -isomorphism from K to itself. Normal extensions Definition 3.11 An extension K of a field F is a normal extension if every irreducible polynomial over F that has at least one zero in K splits over K. Theorem 3.13 A finite extension K of a field F is a normal extensions if and only if K is the splitting field of some polynomial over F. 2

14 MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout IV: Separability; the Theorem of the Primitive Element 4 Separability Definition 4.1 Let f(x) be an irreducible polynomial over a field F. We say that f(x) is separable over F if it has no multiple roots in a splitting field. Definition 4.2 Let f(x) = a 0 + a 1 X + a 2 X a n X n be a polynomial over some field F. The formal derivative of f(x) is the polynomial Df(X) = a 1 + 2a 2 X + 3a 3 X na n X n 1. Lemma 4.4 (Basic properties of formal differentiation) Let f(x) and g(x) be polynomials in F [X] and α and β be scalars in F. Then D ( α f(x) + β g(x) ) = α Df(X) + β Dg(X) D ( f(x) g(x) ) = f(x) Dg(X) + Df(X) g(x) Lemma 4.5 Let f(x) be a polynomial over a field F. Then f(x) has a repeated root in a splitting field if and only if f(x) and the formal derivative Df(X) have a common factor of degree at least one. Proposition 4.6 Let f(x) be an irreducible polynomial over a field F of characteristic zero. Then f(x) is separable. 1

15 Example 4.7 Let t be an indeterminate and consider the field F = F p (t) of rational functions over the finite field F p in the indeterminate t. Define f(x) = X p t, a polynomial in the indeterminate X with coefficients in the field F. One can establish the following facts: (i) f(x) = 0 has no roots in F ; (ii) if α is any root of f(x) in a splitting field, then f(x) = (X α) p ; (iii) f(x) is irreducible over F. Hence f(x) is an inseparable polynomial over the field F. [See Problem Sheet IV, Question 5, for details.] Separable extensions and the Theorem of the Primitive Element Definition 4.8 Let K be an algebraic extension of a field F. We say that K is a separable extension of F if the minimum polynomial of every element of K over F is separable over F. Corollary 4.9 Every algebraic extension of a field of characteristic zero is a separable extension. Lemma 4.10 Let L be a separable extension of an infinite field F and let β, γ L. Then there exists some α F (β, γ) such that F (β, γ) = F (α). Theorem 4.11 (Theorem of the Primitive Element) Let K be a finite separable extension of an infinite field F. Then K = F (α) for some α K. Corollary 4.12 Every finite extension of a field of characteristic zero can be expressed as a simple extension. 2

16 MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout V: Finite Fields 5 Finite Fields Construction of finite fields Proposition 5.1 A finite field F has order p n where p is a prime number equal to the characteristic of F and where n is the degree of F over its prime subfield F p. Lemma 5.2 Let F be a finite field of order q = p n and characteristic p. Then (i) a q 1 = 1 for all a F \ {0}; (ii) ( Freshman s Exponentiation ) for all a, b F and non-negative integers k. (a + b) pk = a pk + b pk Theorem 5.3 Let p be a prime number and n be a positive integer. Then there is precisely one field of order p n up to isomorphism. Definition 5.4 The (unique) field of order p n is denoted F p n and is often called the Galois field of order p n. The multiplicative group of a finite field Definition 5.6 The exponent of a finite group is the least common multiple of the orders of elements of G. Lemma 5.7 Let G be a finite abelian group with exponent ν. Then there exists some g G of order ν. Theorem 5.8 The multiplicative group of a finite field is cyclic. Corollary 5.9 Let F K be an extension of finite fields. Then K = F (α) for some α K. 1

17 MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout VI: Galois groups; the Fundamental Theorem of Galois Theory 6 Galois Groups and the Fundamental Theorem of Galois Theory Galois groups Definition 6.1 Let K be an extension of the field F. The Galois group Gal(K/F ) of K over F is the set of all F -automorphisms of K with binary operation being composition of automorphisms. The sets F and G Definition 6.3 Let K be an extension of the field F and let G = Gal(K/F ) be the Galois group of K over F. (i) Define G to be the set of subgroups of G. (ii) Define F to be the set of intermediate fields; that is, (iii) If H G, define F = { L L is a field with F L K }. H = { x K xφ = x for all φ H }, the set of points in K fixed by all F -automorphisms in H. (iv) If L F, define L = { φ G xφ = x for all x L }, the set of all F -automorphisms that fix all points in L. We shall show that (iii) and (iv) in this definition provide us with maps : G F and : F G and then investigate properties of these maps. Lemma 6.4 Let K be an extension of the field F and G = Gal(K/F ). (i) If H G, then H F ; (ii) If L F, then L G ; (iii) If H 1, H 2 G with H 1 H 2, then H 1 H 2 ; (iv) If L 1, L 2 F with L 1 L 2, then L 1 L 2. Thus our definitions of provide us with maps G F and F G that reverse inclusions. 1

18 The Fundamental Theorem of Galois Theory Definition 6.5 A finite extension of fields is called a Galois extension if it is normal and separable. Lemma 6.6 Let K be a finite Galois extension of a field F and L be an intermediate field (F L K). Then K is a Galois extension of L. Theorem 6.7 (Fundamental Theorem of Galois Theory) Let K be a finite Galois extension of a field F and G = Gal(K/F ). Then: (i) G = K : F. (ii) The maps H H and L L are mutual inverses and give a one-one inclusion-reversing correspondence between G and F. (iii) If L is an intermediate field, then K : L = L and L : F = G / L. (iv) An intermediate field L is a normal extension of F if and only if L is a normal subgroup of G. Moreover, in this situation, Gal(L/F ) = G/L. Tools used in the proof of the Fundamental Theorem Lemma 6.8 Let K be a finite Galois extension of a field F and G = Gal(K/F ). The fixed field of G, G = Fix K (G) = { x K xφ = x for all φ G }, is precisely the base field of F. Lemma 6.9 Let K be a finite separable extension of a field F and let H be a finite group of F -automorphisms of K (that is, H is some subgroup of Gal(K/F )). Then (where H = Fix K (H)). K : H = H 2

19 Lemma 6.10 Let K be a finite Galois extension of a field F and G = Gal(K/F ). The following conditions on an intermediate field L are equivalent: (i) L is a normal subgroup of G; (ii) Lφ L for all φ G; (iii) L is a normal extension of F. Definition 6.11 When K is a finite Galois extension of the field F, the maps H H and L L are called the Galois correspondence between the set G of subgroups of the Galois group and the set F of intermediate fields. Final observations for examples of Galois groups Definition 6.12 Let f(x) be a polynomial over a field F. The Galois group Gal(f(X)) of f(x) is the Galois group Gal(K/F ) of the splitting field K of f(x) over F. Lemma 6.14 Let f(x) be a polynomial over the field F, let K be the splitting field of f(x) over F and let Ω be the set of roots of f(x) in K. Then Gal(K/F ) is isomorphic to the group of permutations that it induces on Ω. Since a polynomial of degree n has at most n roots in a splitting field, the above lemma has the following consequence as an immediate corollary. Corollary 6.15 Let f(x) be a polynomial of degree n over a field F. The Galois group of f(x) over F is isomorphic to a subgroup of the symmetric group S n of degree n. Galois groups of finite fields Definition 6.17 The Frobenius automorphism γ of the finite field F p n γ : F p n F p n given by γ : a a p for all a F p n. of order p n is the map Lemma 6.18 The Frobenius automorphism γ of F p n, given by aγ = a p for all a F p n, is an F p -automorphism of F p n (that is, an element of the Galois group Gal(F p n/f p )). Theorem 6.19 Let p be a prime number and n a positive integer. Then the Galois group Gal(F p n/f p ) of the Galois field of order p n over its prime subfield is cyclic of order n generated by the Frobenius automorphism γ. 3

21 Soluble groups and other group theory Definition 7.6 A group G is called soluble (solvable in the U.S.) if there are subgroups G = G 0 G 1 G 2... G d = 1 (1) such that, for each i = 1, 2,..., d, the subgroup G i is normal in G i 1 and the quotient group G i 1 /G i is abelian. Basic Observations: An abelian group is soluble. A non-abelian simple group is not soluble. Proposition 7.7 (i) If G is soluble, then every subgroup of G is soluble. (ii) If G is soluble, then every quotient group of G is soluble. (iii) If N is a normal subgroup of G such that G/N and N are both soluble, then G is soluble. Proposition 7.8 Let G be a finite soluble group. Then G has a chain of subgroups G = H 0 > H 1 > H 2 > > H n = 1 such that, for i = 1, 2,..., n, H i is a normal subgroup of H i 1 and H i 1 /H i is cyclic of prime order. Theorem 7.9 (Cauchy s Theorem) Let G be a finite group and p be a prime number that divides the order of G. Then G contains an element of order p. Examples of polynomials with abelian Galois groups Lemma 7.10 Let F be a field of characteristic zero and let K be the splitting field of X p 1 over F, where p is a prime number. Then the Galois group Gal(K/F ) is abelian. Lemma 7.11 Let F be a field of characteristic zero in which X n 1 splits. Let λ F and let K be the splitting field for X n λ over F. Then the Galois group Gal(K/F ) is abelian. 2

22 Galois groups of normal radical extensions Theorem 7.12 Let F be a field of characteristic zero and K be a normal radical extension of F. Then the Galois group Gal(K/F ) is soluble. Corollary 7.13 (Galois) Let f(x) be a polynomial over a field F of characteristic zero. If f(x) is soluble by radicals then the Galois group of f(x) over F is soluble. A polynomial which is insoluble by radicals Lemma 7.14 Let p be a prime and f(x) be an irreducible polynomial of degree p over Q. Suppose that f(x) has precisely two non-real roots in C. Then the Galois group of f(x) over Q is isomorphic to the symmetric group S p. Example 7.15 The quintic polynomial f(x) = X 5 9X + 3 over Q is not soluble by radicals. Galois s Great Theorem Lemma 7.16 Let K be a finite normal extension of a field F of characteristic zero and suppose that X p 1 splits in F (for some prime p). If Gal(K/F ) is cyclic of order p then K = F (α) for some α satisfying α p F. Theorem 7.17 (Galois s Great Theorem) Let f(x) be a polynomial over a field F of characteristic zero. Then f(x) is soluble by radicals if and only if the Galois group of f(x) over F is soluble. 3

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