Questions and Answers of Algebra Graduate Texts In Mathematics

Let S be a subset of F. If u ϵ F is algebraic over K(S) and u is not algebraic over K(S - { v }), where v ϵ S, then v is algebraic over K((S - {vi) ∪ {u}).
Proof that Let char K = p ≠ 0 and assume F = K(u,v) where uP ϵ K, vP EK and [F: K] = p2. Then F is not a simple extension of K. Exhibit an infinite number of intermediate fields.
Assume that ∫(x) ϵ K[x] has distinct roots u1,u2, ... , Un in the splitting field F and let G = AutKF n be the Galois group of ∫. Let Y1, . .. , Yn be indeterminates and define:(a) Show
Let K be a subfield of the real numbers and ∫ ϵ K[x] an irreducible quartic. If f has exactly two real roots, the Galois group off is S4 or D4.
Determine all the subgroups of the Galois group and all of the intermediate fields of the splitting field (over Q) of the polynomial (x3 - 2)(x2 - 3) ϵ Q[x]
(a) Let p be a prime and assume either (i) char K = p or (ii) char K ≠ p and K contains a primitive pth root of unity. Then xP - a ϵ K[x] is either irreducible or splits in K[x].(b) If char K = p
(a) Give an example of a field extension K ⊂ F such that u,v ϵ F are transcendental over K, but K(u,v) ≅ K(x1,x2). (b) State and prove a generalization of Theorem 1.5 to the case of n
A finite division ring Dis a field. Here is an outline of the proof (in which E* denotes the multiplicative group of nonzero elements of a division ring E).(a) The center K of Dis a field and D is a
Which roots of unity are contained in the following fields: Q(i), Q(√2), Q(√3),
If n > 2 and ζ is a primitive nth root of unity over Q, then [Q(ζ+ ζ-1) : Q] = φ(n)/2.
Let Fn be a cyclotomic extension of Q of order n.(a) Determine AutQF5 and all intermediate fields.(b) Do the same for F8.(c) Do the same for F7; if ζ is a primitive 7th root of unity what is the
Let Fn be a cyclotomic extension of Q of order n. Determine the structure of AutQFn for every n.
Establish the following properties of the Euler function φ, (a) If p is prime and n > 0, then φ(pn) = pn(1- 1/p) = pn(1-1/p).(b) If (m, n) = 1, then φ(mn) = φ(m)φ(n). (c) If n = p1k1. •
If F is a radical extension field of K and E is an intermediate field, then F is a radical extension of E.
AutQR is the identity group. Since every positive element of R is a square, it follows that an automorphism of R sends positives to positives and hence that it preserves the order in R. Trap a given
Let F be a finite dimensional extension of a finite field K. The norm NKF and the trace TKF(considered as maps F → K) are surjective.
Let φ be the Euler function. (a) φ(n) is even for n > 2.(b) Find all n > 0 such that (c) Find all pairs (n, p) (where n, p > 0, and p is prime) such that φ(n) = n/p.
Let K be a field, ∫ ϵ K[x] an irreducible polynomial of degree n ≥ 5 and F a splitting field of ∫ over K. Assume that AutKF ≅ Sn. Let u be a root of ∫ in F. Then(a) K(u) is not Galois over
If f(x) = let f(x5) be the polynomialEstablish the following i=O i=O properties of the cyclotomic polynomials gn(x) over Q.(a) If p is prime and k ≥ 1, then gpk(x) = gp(xPk-1).(b) If n = p1r1•
Let Q̅ be a (fixed) algebraic closure of Q and τ ϵ Q̅, τ ∉ Q. Let E be a subfield of Q̅ maximal with respect to the condition τ ϵ Q̅. Prove that every finite dimensional extension of E is
(a) If p is an odd prime and n > 0, then the multiplicative group of units in the ring Zpn is cyclic of order pn-1(p - 1).(b) Part (a) is also true if p = 2 and 1 ≤ n ≤ 2.(c) If n ≥ 3, then
Let K be a field, K̅ an algebraic closure of K and σ ϵ AutKK̅. Let Then F is a field and every finite dimensional extension of F is cyclic. F = {ue Ko(u)
Proof that If F is a radical extension field of E and E is a radical extension field of K, then F is a radical extension of K.
Let K be a field with char K ≠ 2,3 and consider the cubic equation x3+a1x2+ a2x+ a3 = 0 (ai ϵ K). Let p = (with cube roots chosen properly). Then the solutions of the given equation are P + Q -
Proof that If F is a cyclic extension of K of degree pn (p prime) and L is an intermediate field such that F = L(u) and L is cyclic over K of degree pn- 1, then F = K(u).
Calculate the nth cyclotomic polynomials over Q for all positive n with n ≤ 20.
If char K = p ≠ 0, let Kp= {up - u| u ϵ K}.(a) A cyclic extension field F of K of degree p exists if and only if K ≠ KP.(b) If there exists a cyclic extension of degree p of K, then there exists
Let F, K, S, P be as in Theorem 6.7 and suppose E is an intermediate field. Then(a) F is purely inseparable over E if and only if S ⊂ E.(b) If F is separable over E, then P ⊂ E.(c) If E ∩ S =
For any homomorphism f : A - B of left R-modules the diagram is commutative, where θA,θB are as in Theorem 4.12 and ∫* is the map induced on A** = HomR(HomR(A,R),R) by the map ∫̅: HomR(B,R) -
The group G contains a subgroup P of order 7 such that the smallest normal subgroup of G containing P is G itself. ex = x and (gig₂)x gi(gzx).
Let R be a commutative ring with identity and Then /is a 1=0 unit in R[x] if and only if ao is a unit in Rand a1, ... , an are nilpotent elements of R η f=Σ Σ aix* e R[x]. 1=0
Let be a free Z-module with an infinite basis X. Then {∫x|x ϵ X} .nX (Theorem 4.11) does not form a basis of F*. Data from theorem 4.11(i) If fx1rl + fx2r2+ ... + fxnrn = 0 (ri e R; Xi eX), then
If (1) ≠ N ⊲ G, then N = G; hence G is simple. [Use Exercise 18 to show P < N; apply Exercise 19.]Data from exercise 18If (1) ≠ N ⊲ 1 G, then 7 divides |N|.Data from exercise 19The group G
Proof that Let S be a multiplicative subset of an integral domain R such that O ∉ S. If R is a principal ideal domain [resp. unique factorization domain], then so is s-1R.
Let R be a ring with more than one element such that for each nonzero a ϵ R there is a unique b ϵ R such that aba = a. Prove:(a) R has no zero divisors.(b) bab = b.(c) R has an identity.(d) R is a
Let R be a ring without identity and with no zero divisors. Let S be the ring whose additive group is R X Z as in the proof of Theorem 1.10. Let A= {(r,n) ϵ S|rx+nx = O for every x ϵ R}.(a) A is an
Let R be a ring without identity. Embed Rina ring S with identity and characteristic zero as in the proof of Identify R with its image in S.(a) Show that every element of S may be uniquely expressed
Proof that(a) [F : K] = 1 if and only if F = K.(b) If [F: K] is prime, then there are no intermediate fields between F and K.(c) If u ϵ F has degree n over K, then n divides [F: K].
Let char K = p ≠ 0 and let n ≥ 1 be an integer such that (p,n) = 1. If v ϵ F and nv ϵ K, then v ϵ K.
If 0 ≠ d ϵ Q, then AutQis the identity or is isomorphic to Z2, -Q(√d)
Suppose ∫ ϵ K[x] splits in F as ∫ = (x - u1)n1 • • • (x - uk)nk (ui distinct; ni ≥ 1). Let v0 , ••• , vk be the coefficients of the polynomial g = (x - u1)(.x - u2) ••• (x -
(a) If F is a field and σ : F → Fa (ring) homomorphism, then σ = 0 or σ is a monomorphism. If σ ≠ 0, then σ(1F) = 1F,(b) The set Aut F of all field automorphisms F → F forms a group under
Proof that F is a splitting field over K of a finite set {∫1 ... , ∫n) of polynomials in K[x] if and only if Fis a splitting field over K of the single polynomial ∫= ∫1∫2 · · ·∫n.
If K is a finite field of characteristic p, describe the structure of the additive group of K.
Give an example of a finitely generated field extension, which is not finite dimensional.
Proof that If u ϵ F is purely inseparable over K, then u is purely inseparable over any intermediate field E. Hence if F is purely inseparable over K, then F is purely inseparable over E.
Suppose K is a subfield of R (so that F may be taken to be a subfield of C) and that /is irreducible of degree 3. Let D be the discriminant off Then (a) D > 0 if and only if /has three real roots.
Let L and M be subfields of F and LM their composite.(a) If K ⊂ L ∩ M and M = K(S) for some S ⊂ M, then LM = L(S).(b) When is it true that LM is the set theoretic union L ∪ M?(c) If E1, ... ,
Proof that If F is a splitting field of S over K and E is an intermediate field, then F is a splitting field of S over E.
What is the Galois group of over Q? Q(√2,√3,√5)
If p ϵ Z is prime, then aP = a for all a ϵ Zp or equivalently, cP = c (mod p) for all c ϵ Z.
Proof that If u1, ... , Un ϵ F then the field K(u1, ... , un) is (isomorphic to) the quotient field of the ring K[u1, ... , Un].
Proof that If F is purely inseparable over an intermediate field E and E is purely inseparable over K, then F is purely inseparable over K.
Let ∫ be a separable cubic with Galois group S3 and roots u1,u2,U3 ϵ F. Then the distinct intermediate fields of the extension of K by F are F, K(Δ), K(u1), K(u2), K(u3), K. The corresponding
(a) Let E be an intermediate field of the extension K ⊂ F and assume that E = K(u1, ... , ur) where the ui, are (some of the) roots of ∫ ϵ K[x]. Then F is a splitting field of ∫ over K if and
Proof that:(a) If O ≠ d ϵ Q, then is Galois over Q. (b) C is Galois over R. (PM)D
Proof that If |K| = pn, then every element of K has a unique pth root in K
(a) For any u1, . . . , un ϵ F and any permutation σ ϵ Sn, K(u1, . . . , Un) = K(uσ(1),. .. , uσ(n))(b) K(u1, ... , Un-1)(un) = K(u1, ... , Un). (c) State and prove the analogues of (a) and
Proof that If u ϵ F is separable over K and v ϵ F is purely inseparable over K, then K(u,v) = K(u + v). If u ≠ 0, v ≠ 0, then K(u,v) = K(uv).
Let ∫/ g ϵ K(x) with ∫/ g ∉ K and ∫,g relatively prime in K[x] and consider the extension of K by K(x). (a) x is algebraic over K(∫/g) and [K(x): K(∫/g)] = max (deg ∫,deg g).(b) If E
Proof that If char K ≠ 2,3 then the discriminant of x3 + bx2 +cx+ d is -4c3 - 27d2 + b2(c2 - 4bd) + 18bcd.
Proof that If F is a splitting field over K of S, then F is also a splitting field over K of the set T of all irreducible factors of polynomials in S.
Proof that If the roots of a monic polynomial ∫ ϵ K[x] (in some splitting field of ∫ over K) are distinct and form a field, then char K = p and ∫ = xPn - x for some n ≥ l.
Proof that If char K = p ≠ 0 and u ϵ K but a ∉ Kp, then x-pn - a ϵ K[x] is irreducible for every n > 1.
If char K ≠ 2 and ∫ ϵ K[x] is a cubic whose discriminant is a square in K, then ∫ is either irreducible or factors completely in K.
Proof that If ∫ ϵ K[x] has degree n and F is a splitting field of ∫ over K, then [F : K] divides n !.
(a) Construct a field with 9 elements and give its addition and multiplication tables.(b) Do the same for a field of 25 elements.
Proof that Every element of K(x1, ... , xn) which is not in K is transcendental over K.
Proof that lf ∫ ϵ K[x] is monic irreducible, deg ∫≥ 2, and ∫ has all its roots equal (in a splitting field), then char K = p ≠ 0 and ∫ = xPn - a for some n ≥ 1 and a ϵ K.
Proof that Over any base field K, x3 - 3x + 1 is either irreducible or splits over K.
Proof that Let K be a field such that for every extension field F the maximal algebraic extension of K contained in F is K itself. Then K is algebraically closed.
Proof that If IKI = q and (n,q) = I and F is a splitting field of xn - IK over K, then [F: K] is the least positive integer k such that n | (cl - I).
Proof that If v is algebraic over K(u) for some u E F and vis transcendental over K, then u is algebraic over K(v).
(a) Consider the extension Q(u) of Q generated by a real root u of x3 - 6x2+ 9x + 3. (Why is this irreducible?) Express each of the following elements in terms of the basis {1,u,u2 } : u4 ;u5
Proof that If G/C(G) is cyclic, then G is abelian.
Any group of order p2q (p,q primes) is solvable.
Show that D4 is not isomorphic to Q8.
Sn is solvable for n ≤ 4, but S3 and S4 are not nilpotent.
Proof that Any finite group is isomorphic to a subgroup of An. for some n.
Proof that There is no group G such that G' = S4.
Proof that Any simple group G of order 60 is isomorphic to A5.
Proof thatThere are no nonabelian simple groups of order < 60.
Proof that The additive group Q is indecomposable.
Let G be the subset of AutKK(x) consisting of the three automorphisms induced (as in 6 (c)) by x|→ x, |→1K/(1K - x), x|→(x - 1K)/x. Then G is a subgroup of AutKK(x). Determine the fixed field
Prove that S4 has no transitive subgroup of order 6.
If n is an odd integer such that K contains a primitive nth root of unity and char K I- 2, then K also contains a primitive 2nth root of unity.
If F is algebraically closed and E consists of all elements in F that are algebraic over K, then E is an algebraic closure of K
Proof that If IKI = q and ∫ ϵ K[x] is irreducible, then ∫ divides xqn - x if and only if deg ∫ divides n.
Proof that If u ϵ F is algebraic of odd degree over K, then so is u2 and K(u) = K(u2).
Proof that If char K = p ≠ 0 and [F: K] is finite and not divisible by p, then F is separable over K.
Let ∫ be an (irreducible) separable quartic over K and u a root of ∫. There is no field properly between K and K(u) if and only if the Galois group of ∫ is either A4 or S4.
Assume char K = 0 and let G be the subgroup of AutKK(x) that is generated by the automorphism induced by x I→ x + 1K. Then G is an infinite cyclic group. Determine the fixed field E of G. What is
If |K| = pr and IFI = pn, then r I n and AutKF is cyclic with generator φ given by U |→uPr.
If xn - a ϵ K[x] is irreducible and u ϵ F is a root of xn - a and m divides n, then prove that the degree of um over K is n/m. What is the irreducible polynomial for um over K?
Let char K = p ≠ 0. Then an algebraic element u ϵ F is separable over K if and only if K(u) = K(uPn) for all n ≥ 1.
If d ≥ 0 is an integer that is not a square describe the field and find a set of elements that generate the whole field. -Q(√d)
Let x4 + ax2 +b ϵ K[x] (with char K ≠ 2) be irreducible with Galois group G.(a) If b is a square in K, then G = V.(b) If b is not a square in K and b(a2 - 4b) is a square in K, then G ≅ Z4.(c)
Proof that(a) If K is an infinite field, then K(x) is Galois over K. (b) If K is finite, then K(x) is not Galois over K.
F is an algebraic closure of K if and only if F is algebraic over K and for every algebraic extension E of K there exists a K-monomorphism E → F.
Proof that If n ≥ 3, then x2n + x + 1 is reducible over z2 •

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