MAT 535 Problem Set 5 Solutions


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1 Final Exam, Tues 5/11, :15pm4:45pm Spring 010 MAT 535 Problem Set 5 Solutions Selected Problems (1) Exercise 9, p 617 Determine the Galois group of the splitting field E over F = Q of the polynomial f(x) = x 4 + 4x 1 () Exercise 19, p 618 Describe the Galois group of an irreducible quartic polynomial over Q with negative discriminant (3) Exercise 48, p 63 Determine the splitting field E and Galois group of the sextic polynomial f(x) = x 6 x 3 over Q (4) Exercise 19, p 638 Prove that a quadratic extension Q( D)/Q is contained in a degree 4 Galois extension of Q with cyclic Galois group if and only if D is a sum of two rational squares (5) Exercise 13, p 853 Prove that for simple, left Rmodules M and N (R is a unital associative ring), every nonzero Rmodule homomorphism from M to N is an isomorphism Conclude that Hom R (M, M) is a division ring Solutions to Selected Problems Solution to (1) By Proposition 11 on p 308, if f(x) has a root then the root is ±1 But f( 1) equals 4 and f(1) equals 4 So if f(x) factors, it is a product of two irreducible, monic, quadratic polynomials By Gauss s lemma, the factors are in Z[x] And by considering the constant coefficient and the coefficient of x 3, the factorization must be (x + (ax + 1))(x (ax + 1)) = x 4 a x ax 1 But since the coefficient of x in f(x) is 0, there is no such factorization Thus f(x) is irreducible The resolvent cubic of f(x) is Notice that this factors, h(z) = z 3 + 4z + 16 h(z) = (z + )(z z + 8) = (z ( ))k(z), k(z) = z z + 8 = (z 1) + 7 Thus the discriminant of h(z) is Disc(h) = [k( )] Disc(k) = [ 4 ] ( 8) = 6 7 1
2 Final Exam, Tues 5/11, :15pm4:45pm Spring 010 And the discriminant D of f(x) equals the discriminant of the resolvent cubic h(z) Observe that this is not a square Up to square factors, the discriminant is congruent to 7 Thus Q( D) equals Q( 7) The claim is that f(x) is irreducible in Q( 7) Denote by t one of the two square roots of 7 Then the Galois group is Aut(Q( 7)/Q) = {1, σ}, where σ(t) equals t Since f(x) is irreducible of degree 4 over Q, every root generates a field extension of degree 4 Since [Q( 7) : Q] equals, f(x) has no roots in Q( 7) So the only possible factors are irreducible quadratic factors If f(x) has an irreducible quadratic factor, then it has a monic irreducible quadratic factor, say q(x) Since f(x) is defined over Q, then also σ(q(x)) is an irreducible quadratic factor Since Q( 7)[x] is a UFD, it follows that f(x) = q(x)σ(q(x)) In particular, f(0) equals q(0)σ(q(0)) By direct computation we have, q(0) = a + bt, σ(q(0)) = a bt, q(0)σ(q(0)) = a b t = a + 7b for some elements a, b in Q Thus a + 7b equals f(0) = 1 But a and 7b are nonnegative, thus they cannot sum to the negative number 1 This contradiction proves that f(x) is irreducible in Q( 7) Thus, by the discussion on p 615, the Galois group of E over Q is isomorphic to a conjugate of the dihedral group D 8 as a subgroup of symmetric group S 4 Solution to () For every monic polynomial f(x) of degree d, in a splitting field K of f(x) the discriminant D equals ( ) (α i α j ) 1 i<j d where f(x) = (x α 1 ) (x α d ) Thus D is a square in K So K contains Q( D) Now let K be any Galois extension of Q By the fundamental theorem of algebra, there exists an embedding of K into C, say φ : K C Moreover, for every embedding φ : K C, there exists a unique element σ of Aut(K/Q) such that φ equals φ σ In particular, for complex conjugation τ : C C, τ φ equals φ τ K for some element τ K in Aut(K/Q) Since τ (τ φ) equals Id C φ equals φ, also (φ τ K ) τ K equals φ, ie, τ K τ K equals Id K Thus τ K is an element of Aut(K/Q) of order dividing, ie, of order 1 or order Also observe that since φ is only welldefined up to φ = φ σ, also τ K is only welldefined up to conjugation στ K σ 1 Moreover, τ K is the identity only if τ φ equals φ, ie, if and only if φ(k) is fixed by τ Since the fixed field of τ is R, τ K equals Id if and only if φ(k) is contained in R for one, and hence every, embedding of K into R Thus τ K is an element welldefined up to conjugation, which equals the identity if K is contained in R, and which has order if K is not contained in R Now assume that f(x) is an irreducible polynomial of degree 4 over Q And assume that the discriminant D is negative Then D is not contained in R Since Q( D) is contained in the splitting field K of f(x), K is not contained in R Thus τ K is an element of Aut(K/Q) of order Moreover, Q( D) is Galois over Q of order Thus Aut(Q( D)/Q) is the quotient of Aut(K/Q) by the normal subgroup Aut(K/Q( D)) This quotient group has order, thus is isomorphic to Z/Z
3 Final Exam, Tues 5/11, :15pm4:45pm Spring 010 And clearly τ K restricts to a nontrivial element of order in Aut(Q( D)/Q), namely complex conjugation Thus τ K is an element of Aut(K/Q) of order whose image in Aut(Q( D)/Q) also has order But the group Z/4Z has a unique surjective homomorphism onto Z/Z, and the kernel is precisely the set of elements of order 1 and Since τ K has order and is not contained in the kernel of the quotient map, Aut(K/Q) cannot be isomorphic to Z/4Z A bit more generally, let D be a negative element of Q, and let K be a field containing Q( D) such that K/Q is Galois of degree 4 Then, as above, τ K is an element of order in Aut(K/Q) which maps to an element of order in the quotient Aut(Q( D)/Q) Thus, for the same reason as above, Aut(K/Q) is not cyclic of order 4 Solution to (3) The polynomial f(x) is irreducible of degree 6 by Eisenstein s criterion Moreover, for each of the roots β of g(y) = y y, the three roots of x 3 β are roots of f(x) Thus the splitting field contains Q(µ 3 ) as well as the splitting field Q( 3) of g(y) Notice that Q(µ 3 ) equals Q( 3) So the splitting field also contains 3/ 3, which is a square root of 1 Thus the splitting field contains Q(i, 3) In this field the roots of g(y) = (y 1) 3 are 1 ± 3 Thus the roots of f(x) are the cube roots of 1 ± 3 Now let α + be a root of f(x) with α 3 + equals Since K contains primitive cube roots of 1, up to multiplication by such we may assume that α + is real Similarly let α be a real root of f(x) with α 3 equals 1 3 Then ( α + α ) 3 equals ( 1) 3 α 3 +α 3 equals ( 1)(1 + 3)(1 3) = Thus α + α is a real cube root of, ie, α + α equals Thus the splitting field E of f(x) contains Q(, i, 3) The polynomial t 3 is irreducible of degree 3 over Q by Eisenstein s criterion So every root generates an extension of Q of degree 3 And Q(i, 3) is a biquadratic extension of Q of degree 4 Thus Q(i, 3) contains no root of t 3 Since t 3 is a cubic polynomial with no root in Q(i, 3), it is irreducible over Q(i, 3) Thus the root field Q(, i, 3) is degree 3 over Q(i, 3) So [Q(, i, 3) : Q] equals 3 4 = 1 Denote by L the field Q(, i, 3) Then α 3 + equals 1+ 3, which is an element of L Since L contains a primitive cube root of 1, L(α + ) contains every cube root of 1 + 3, hence it is the splitting field of t 3 (1 + 3) So [L(α + ) : L] equals 3 if there is no cube root of in L, and it equals 1 otherwise Moreover, α equals /α +, so L(α + ) contains L(α ) And then it contains all the roots ζ m 3 α + and ζ m 3 α of f(x), where ζ 3 is a primitive cube root of 1 and where m = 0, 1, So L(α + ) equals the splitting field E So [E : L] equals 3 if L contains no cube root of 1 + 3, and [E : L] equals 1 if L contains a cube root α + of So [E : Q] equals 36 or 1 Now consider the element in K, Observe that α = α + + α α +α = 1, ( α + ) 3 = (1 + 3) = + 3, ( α ) 3 = (1 3) = 3 3
4 Final Exam, Tues 5/11, :15pm4:45pm Spring 010 Thus, we have α 3 = [ ( ) α 3 ( ) + α 3 ] [ ] α +α α + + α = 4 + 3α Therefore α is a root of the cubic polynomial k(x) = x 3 3x 4 By Proposition 11 on p 308, the only possible roots of k(x) in Q are ±1, ±, ±4 By direct computation, none of these are roots Therefore k(x) is irreducible over Q By direct computation, the discriminant equals D = 4( 3) 3 7( 4) = 4 7(1 4) = 3 4 So the splitting field contains the square root D, and thus also D/18 which equals 1 On the other hand, consider the extension L/Q This is a Galois extension, the compositum of the splitting field Q( 3, ) of t 3, which has Galois group S 3, and the quadratic extension Q(i), which has Galois group Z/Z By Corollary on p 593, the Galois group Aut(L/Q) equals Aut(Q( 3, )/Q) Aut(Q(i)/Q) = S 3 Z/Z This has 3 subgroups of index 3, ie, Sylow subgroups By the Fundamental Theorem of Galois Theory, L/Q contains 3 subextensions which have degree 3 over Q But each of the three cube roots of generates a distinct degree 3 extension of Q Thus these are the only degree 3 subextensions of Q And for each such degree 3 subextension E/Q of L/Q, the Galois closure of E/Q is simply the splitting field of t 3, ie, E( 3) In particular, this does not contain i, since L = E( 3, i) has degree 4 over E, not So Q(α) cannot be one of the degree 3 subextensions E/Q of L/Q, since the Galois closure of Q(α) does contain i Since Q(α)/Q is a degree 3 extension, it follows that Q(α) is not contained in L Therefore α is not contained in L Since α is contained in K, K properly contains L Therefore K/L is a degree 3 extension and [K : Q] equals 36 By the argument above, K is the compositum of the two Galois extensions Q( 3, )/Q and Q(α, i)/q, each of which is a degree 6 extension with Galois group S 3 By Corollary 0 on p 59, it follows that these two extensions are linearly disjoint over Q, ie, their intersection equals Q And then by Corollary on p 593, the Galois group of the compositum equals the product of the Galois groups, Aut(K/Q) = Aut(Q( 3, )/Q) Aut(Q(α, i)/q) = S 3 S 3 To be explicit, Aut(Q( 3, )/Q) acts as the full symmetric group S 3 on the three roots,, 1 + 3, 1 3 of t 3 And Aut(Q(α, i)/q) acts as the full symmetric group S 3 on the three roots, of x 3 3x 4 α, α + α α, i, α α α i Nota Bene If char(e), 3, then for a monic cubic polynomial h(x) = x 3 +px +q in E[x] which has discriminant D = 4p 3 7q a square δ in E and which has one root α in E, then the other 4
5 Final Exam, Tues 5/11, :15pm4:45pm Spring 010 two roots are in E and given by the formula α 6pα 9qα + 4p δ Solution to (4) Suppose first that D is a sum of two rational squares, D = s + t Since D is squarefree, both s and t are nonzero Thus ± D ± s D are four distinct elements And by direct computation, these are roots of the monic, quartic polynomial f(x) = x 4 Dx + t D = (x D) s D Each of these roots generates an extension of Q which contains Q( D) So the splitting field of f(x) contains Q( D), and none of the roots of f(x) lies in Q Thus if f(x) is reducible, it is a product of two irreducible quadratic factors Since f(x) has no cubic or linear term, the factorization would have to be f(x) = (x + ax + b)(x ax + b) = x 4 + (b a )x + b But then t D equals b, so that D is the rational square (b/t) This contradicts that D is squarefree Therefore f(x) is an irreducible quartic polynomial Let α + be a root of f(x) with Now (D + s D)(D s D) = t D so that Therefore, defining α + = D + s D D s D = ( t ) D α + α = t D = s + D α + t it follows that Q(α + ) contains a root α of f(x) with α = D s D Thus ±α + and ±α are the four roots of f(x) So Q(α + ) is a splitting field of f(x) So Q(α + )/Q is a Galois extension of degree 4 Let σ : Q(α + ) Q(α + ) be an automorphism with σ(α + ) equals α Then σ(α +) equals σ(α + ), which equals (α ) = D s D But α + equals D + s D Thus σ( D) equals D And thus we have σ(α ) = tσ( D) σ(α + ) α + = t( D) α = α + In particular, σ(σ(α + )) equals α + Thus σ does not equal the identity Since the order of σ divides #Aut(Q(α + )/Q) = [Q(α + ) : Q] = 4, it follows that σ is an element of order 4 Thus 5
6 Final Exam, Tues 5/11, :15pm4:45pm Spring 010 Aut(Q(α + )/Q) is a cyclic group of order 4 Therefore, if D is a squarefree integer which is a sum of two rational squares, then Q( D) is contained in a degree 4, Galois extension of Q with cyclic Galois group Conversely, assume that K = Q( D) is contained in a degree 4, Galois extension L/Q with cyclic Galois group In particular, L/K is a degree extension By Kummer s Theorem, Proposition 37 on p 66, L/K is of the form L = K( u) for some element u = a + b D in K Since K/Q is a Galois extension, Aut(L/K) is a normal subgroup of Aut(L/Q), and the quotient is Aut(K/Q) In particular, there exists an element σ of Aut(L/Q) which maps to the nontrivial element of Aut(K/Q) And then we have (σ( u)) = σ( u ) = σ(u) = a b D Therefore v := uσ( u) is an element of L whose square equals v = u (σ( u)) = (a + b D)(a b D) = a b D Thus Q(v)/Q is a degree subextension of L/Q Since L/Q is Galois, degree subextensions are in bijection with index subgroups of Aut(L/Q) Since Aut(L/Q) is a cyclic group of order 4, it has a unique index subgroup Thus the degree subextension Q(v) must equal Q( D) So v is an element of Q( D) with w := v an element in Q By another part of Kummer s theorem, for every element v in Q( D) with w := v in Q, then either w = c or w = c D for some c in Q Now if w equals c, then v = ±c is in Q But then σ(u) = v/u implies that σ(σ(u)) equals v/σ(u) = u So σ fixes the generator u of Q(u) So σ is the identity, ie, the order of σ divides But since Aut(L/Q) is a cyclic group of order 4, and since σ is not in Aut(L/K), the unique subgroup of index, σ is a generator, ie, σ has order 4 This contradiction proves that w is not of the form c Therefore w is for the form c D And then since w := v also equals a b D, this implies the identity, D = ( ) bd + a ( ) cd a Therefore D is a sum of two rational squares Thus the quadratic extension Q( D)/Q is contained in a degree 4, Galois extension of Q with cyclic Galois group if and only if D is a sum of two rational squares In particular, by the same argument as in the solution of Exercise 10, p 58 on the previous problem set, 3 is not a sum of two rational squares Thus Q( 3) is not contained in a degree 4, Galois extension of Q with cyclic Galois group Similarly, since a negative real number cannot be a sum of two squares, for rational D < 0, the extension Q( D)/Q is not contained in a degree 4, Galois extension of Q with cyclic Galois group Solution to (5) Let f : M N be a nonzero Rmodule homomorphism Then Ker(f) is a left Rsubmodule of M which does not equal M Since M is simple, the only proper left Rsubmodule 6
7 Final Exam, Tues 5/11, :15pm4:45pm Spring 010 of M is {0} Thus f is injective And Image(f) is a submodule of N which is isomorphic to M By definition, the zero module is not simple, thus M is nonzero, and so also Image(f) is nonzero Since N is simple, the only nonzero left Rsubmodule of N is all of N Thus Image(f) equals N Thus f is surjective Since f is injective and surjective, f is an isomorphism of left Rmodules Now consider the case when N = M is a left Rmodule Then Hom R (M, M) is a ring under the usual addition, and with composition for multiplication The identity map is the multiplicative unit And it is associative since composition is associative Therefore Hom R (M, M) is a unital, associative ring Also for the center Z(R) of R, scaling by elements in Z(R) gives a ring homomorphism Z(R) Hom R (M, M) which makes Hom R (M, M) into a Z(R)algebra, ie, the image of Z(R) is in the center of the ring Hom R (M, M) (but the center of Hom R (M, M) might be strictly larger than the center of R) Finally, if M is simple, then the argument above proves that every nonzero element of Hom R (M, M) is invertible in this ring Thus Hom R (M, M) is a division ring As one motivating example, when Q 8 is the quaternion group, when R equals R Q 8 is the group algebra of Q 8 over R, and when M is the simple Rmodule described in Example 8 on p 845, then the division ring Hom R (M, M) is the usual quaternion algebra H 7
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