Questions and Answers of Algebra Graduate Texts In Mathematics

Proof that If F is algebraic over K and D is an integral domain such that K ⊂ D ⊂ F, then D is a field.
Let char K = p ≠ 0 and let ∫ ϵ K[x] be irreducible of degree n. Let m be the largest nonnegative integer such that ∫ is a polynomial in xPm but is not a polynomial in xpm+1. Then n = nopm. If
If K is an infinite field, then the only closed subgroups of AutKK(x) are itself and its finite subgroups.
Proof that F is an algebraic closure of K if and only if F is algebraic over K and for every algebraic field extension E of another field K1 and isomorphism of fields σ : K1 → K, σ extends to a
(a) If find [F: Q] and a basis of F over Q. (b) Do the same for where i ϵ C, i2 = -1, and ω is a complex (nonreal) cube root of 1. F = Q(√√²,√√³),
Every element in a finite field may be written as the sum of two squares.
Proof that If ∫ ϵ K[x] is irreducible of degree m > 0, and char K does not divide m, then ∫ is separable.
In the extension of Q by Q(x), the intermediate field Q(x2) is closed, but Q(x3) is not.
(a) If u1, ... , Un E Fare separable over. K, then K(u1, ... , un) is a separable extension of K. (b) If F is generated by a (possibly infinite) set of separable elements over K, then F is a
Let F be an algebraic closure of ZP (p prime).(a) F is algebraic Galois over ZP.(b) The map φ: F→ F given by u| → uP is a nonidentity Zp-automorphism of F. (c) The subgroup H = (φ) is
Proof that F is purely inseparable over K if and only if F is algebraic over K and for any extension field E of F, the only K-monomorphism F →E is the inclusion map.
Proof that If E is an intermediate field of the extension such that E is Galois over K, F is Galois over E, and every σ ϵ AutKE is extendible to F, then F is Galois over K.
Prove that the field C, Q(i) and Q(√2) are isomorphic as vector spaces, but not as fields. Q(√2)
Let E be an intermediate field.(a) If u ϵ F is separable over K, then u is separable over E.(b) If F is separable over K, then F is separable over E and E is separable over K.
Proof that If K is finite and F is an algebraic closure of K, then AutKF is abelian. Every element of AutKF (except 1F) has infinite order.
If F is a finite dimensional Galois extension of K and E is an intermediate field, then there is a unique smallest field L such that E ⊂ L ⊂ F and L is Galois over K; furthermore where σ runs
Let F be an algebraic closure of the field Q of rational numbers and let E ⊂ F be a splitting field over Q of the set S = { x2 + a | a ϵ Q} so that E is algebraic and Galois over Q (Theorem
(a) The following conditions on a field K are equivalent:(i) every irreducible polynomial in K[x] is separable;(ii) every algebraic closure K of K is Galois over K;(iii) every algebraic extension
In the extension of an infinite field K by K(x,y), the intermediate field K(x) is Galois over K, but not stable (relative to K(x,y) and K). [See Exercise 9]Data from exercise 9 (a) If K is an
Suppose [F : K] is finite. Then the following conditions are equivalent:(i) F is Galois over K;(ii) F is separable over K and a splitting field of a polynomial ∫ ϵ K[x];(iii) F is a splitting
If σ ϵ Sn, then the map K(x1, ... 'Xn) - K(x1, ... 'Xn) given by is a K-automorphism of K(x1, •.. , xn). f(x₁,...,xn), f(xo(1), Xo(n)) g(x₁,...,xn) g(xo(1),..., Xo(n)) ...,
If F = K(u,v) with u,v algebraic over Kand u separable over K, then Fis a simple extension of K.
Let F be a finite dimensional Galois extension of K and let L and M be two intermediate fields.(a) AutLMF = AutLF ∩ AutMF;(b) AutL∩MF = AutLF ∨ AutMF;(c) What conclusion can be drawn if AutLF
Part (ii) or (ii)' of the Fundamental Theorem (2.5) is equivalent to: an intermediate field E is normal over K if and only if the corresponding subgroup E' is normal in G = AutKF in which case Gi E'
Here is a method for constructing a polynomial ∫ ϵ Q[x] with Galois group Sn for a given n > 3. It depends on the fact that there exist irreducible polynomials of every degree in Zp[x] (p
Proof that If L and M are intermediate fields such that L is a finite dimensional Galois extension of K, then LM is finite dimensional and Galois over M and AutMLM ≅ AutL∩ML.
In the field K(x), let u = x3/(x + 1). Show that K(x) is a simple extension of the field K(u). What is [K(x) : K(u)]?
Let E be an intermediate field. (a) If F is algebraic Galois over K, then F is algebraic Galois over E. show that the "algebraic" hypothesis is necessary.(b) If F is Galois over E, E is Galois
Proof that Let F be an algebraic extension of K such that every polynomial in K[x] has a root in F. Then F is an algebraic closure of K.
Find an irreducible polynomial f of degree 2 over the field Z2. Adjoin a root u of ∫ toZ2 to obtain a field Z(u) of order 4. Use the same method to construct a field of order 8.
Proof that If an intermediate field E is normal over K, then E is stable (relative to F and K).
A complex number is said to be an algebraic number if it is algebraic over Q and an algebraic integer if it is the root of a manic polynomial in Z[xJ.(a) If u is an algebraic number, there exists an
Let F be normal over K and E an intermediate field. Then E is normal over K if and only if E is stable [see Exercise 17]. Furthermore AutKF/E' ≅ AutKE.Data from exercise 17If an intermediate field
If u,v ϵ F are algebraic over K of degrees m and n respectively, then [K(u,v) : K] ≤ mn. If (m,n) = 1, then [K(u,v) : K] = mn.
Let c,d be constructible real numbers.(a) c + d and c - d are constructible.(b) If d ≠ 0, then c/ d is constructible.(c) cd is constructible.(d) The constructible real numbers form a subfield
Let L and M be intermediate fields in the extension K ⊂ F.(a) [LM: K] is finite if and only if [L : K] and [M: K] are finite.(b) If [LM: K] is finite, then [L : K] and [M: K] divide [LM: K] and
Proof that If F is normal over an intermediate field E and E is normal over K, then F need not be normal over K.
(a) Let L and M be intermediate fields of the extension K ⊂ F, of finite dimension over K. Assume that [LM : K] = [L : K][M : K] and prove that L ∩ M = K.(b) The converse of (a) holds if [L : K]
Proof that Let F be algebraic over K. F is normal over K if and only if for every K-monomorphism of fields σ : F → N, where N is any normal extension of K containing F, σ(F) = F so that σ
Show that F is an algebraic extension of K if and only if for every intermediate field E every monomorphism u : E → E which is the identity on K is in fact an automorphism of E.
Proof that If F is algebraic over K and every element of F belongs to an intermediate field that is normal over K, then F is normal over K.
Proof that If u ϵ F is algebraic over K(X) for some X ⊂ F then there exists a finite subset X' ⊂ X such that u is algebraic over K(X').
Proof that If [F : K] = 2, then F is normal over K.
Let F be a subfield of R and P,Q points in the Euclidean plane whose coordinates lie in F.(a) The straight line through P and Q has an equation of the form ax + by + c = 0, with a,b,c ϵ F.(b) The
An algebraic extension F of K is normal over K if and only if for every irreducible ∫ ϵ K[x]. ∫ factors in F[x] as a product of irreducible factors all of which have the same degree.
Proof that Let F be a splitting field of ∫ ϵ K[x]. Without using Theorem 3.14 show that F is normal over K.Data from theorem 3.14lf F is an algebraic extension field of K, then the following
Proof that Let E1 and E2 be subfields of F and X a subset of F. If every element of E1 is algebraic over E2, then every element of E1(X) is algebraic over E2(X).
Let G be the multiplicative group of all nonsingular 2 X 2 matrices with rational entries. Show that a=has order 4 and b =has order 3, but ab has infinite order. Conversely, show that the additive
No two of D6, A4, and T are isomorphic, where T is the group of order 12 described in Proposition 6.4 and Exercise 5.Data from in exercise 5(a) Show that there is a nonabelian subgroup T of S3 X Z4
Use Exercises 3 and 7 to obtain a proof of Theorem 2.2 which is independent of Theorem 2.1.Data from Exercise 3Suppose G is a finite abelian p-group (Exercise 7) and x ϵ G has maximal order. If Y ϵ
Ifis a free abelian group, and G is the subgroup with basis xaX X' = X - {x0} for some x0 ϵ X, then F/G ≅ Zxo. Generalize this result to arbitrary subsets X' of X. F = Σzx Zx TEX
A (sub)group in which every element has order a power of a fixed prime p is called a p-(sub)group. Let G be an abelian torsion group.(a) G(p) is the unique maximum p-subgroup of G (that is, every p
Let H, K be normal subgroups of a group G such that G = H X K.(a) If N is a normal subgroup of H, then N is normal in G (compare Exercise 1.5.10).(b) If G satisfies the ACC or DCC on normal
Prove that a finite group G is nilpotent if and only if every maximal proper subgroup of G is normal. Conclude that every maximal proper subgroup has prime index.
Proof that If H ⊲ G, where G has a composition series, then G has a composition series one of whose terms is H.
Let G be the multiplicative group generated by the real matricesandIf H is the set of all matrices in G whose (main) diagonal entries are l, then H is a subgroup that is not finitely generated. (12) -
A finite abelian p-group (Exercise 7) is generated by its elements of maximal order.Data from Exercise 7A (sub)group in which every element has order a power of a fixed prime p is called a
Let G be a group and let In G be the set of all inner automorphisms of G. Show that In G is a normal subgroup of Aut G.
Find the Sylow 2-subgroups and Sylow 3-subgroups of S3, S4, S5.
If f and g are endomorphisms of a group G, then ∫ + g need not be an endomorphism.
If G is a nonabelian group of order p 3 (p prime), then the center of G is the sub group generated by all elements of the form aba-1b-1 (a,b ϵ G).
Use the Krull-Schmidt Theorem to prove Theorems 2.2 and 2.6 (iii) for finite abelian groups.Data From Theorem 2.2Every finitely generated abe/ian group G is (isomorphic to) a finite direct sum of
If N is a nontrivial normal subgroup of a nilpotent group G, then N ∩ C(G) ≠ (e).
Proof that A nonzero free abelian group has a subgroup of index n for every positive integer n.
Exhibit an automorphism of Z6 that is not an inner automorphism.
If every Sylow p-subgroup of a finite group G is normal for every prime p, then G is the direct product of its Sylow subgroups.
Let ∫ and g be normal endomorphisms of a group G.(a) ∫g is a normal endomorphism.(b) H ⊲ G implies ∫(H) ⊲ G.(c) If ∫ + g is an endomorphism, then it is normal.
(a) Let G be a finite abelian p-group (Exercise 7). Show that for each n ≥ 0, pn+1G ∩ G[p] is a subgroup of pnG ∩ G[p].(b) Show that (pnG ∩ G[p])/(pn+1G ∩ G[p]) is a direct sum of copies of
Let p be an odd prime. Prove that there are, at most, two nonabelian groups of order p3• [One has generators a,b satisfying |a| = p2 ; |b|= p; b-1ab = ai+P; the other has generators a,b,c
If Dn is the dihedral group with generators a of order n and b of order 2, then(a) a2 ϵ Dn'.(b) If n is odd, Dn' ≅ Zn.(c) If n is even, Dn' ≅ Zm, where 2m = n.(d) Dn is nilpotent if and only if
If H and K are solvable subgroups of G with H ⊲ G.HK is a solvable subgroup of G.
If |G| = pnq, with p > q primes, then G contains a unique normal subgroup of index q.
Proof that Let G = G1 X · · · X Gn. For each i let λi: Gi → G be the inclusion map and πi : G → Gi the canonical projection (see page 59). Let φi,= λiπi,. Then the "sum" φi1 + · · · +
How many subgroups of order p2 does the abelian group Zp3⊕ZP2 have?
Classify up to isomorphism all groups of order 18. Do the same for orders 20 and 30.
Show that the commutator subgroup of S4 is A4. What is the commutator group of A4?
Let G be a finitely generated abelian group in which no element (except 0) has finite order. Then G is a free abelian group.
Show that the center of S4 is (e); conclude that S4 is isomorphic to the group of all inner automorphisms of S4.
Every group of order 12, 28, 56, and 200 must contain a normal Sylow subgroup, and hence is not simple.
A group G is nilpotent if and only if there is a normal series G = Go > G1 > • · · > Gn = (e) such that Gi/Gi+1 < C(G/Gi+1) for every i.
(a) Show that the additive group of rationals Q is not finitely generated.(b) Show that Q is not free.(c) Conclude that Exercise 9 is false if the hypothesis "finitely generated" is omitted.Data from
Let G be a group containing an element a not of order 1 or 2. Show that G has a nonidentity automorphism.
How many elements of order 7 are there in a simple group of order 168?
If G and H are groups such that G X G⊕H X Hand G satisfies both the ACC and DCC on normal subgroups, then G⊕H.
Let G, H, and K be finitely generated abelian groups.(a) If G⊕G ≅ H⊕H, then G ≅ H.(b) If G⊕H ≅ G⊕K, then H ≅ K.(c) If G1 is a free abelian group of rank N0 , then G1⊕ Z⊕Z ≅
Proof that A nontrivial finite solvable group G contains a normal abelian subgroup H ≠ (e). If G is not solvable then G contains a normal subgroup H such that H' = H.
(a) Show that the analogue of Theorem 7.11 is false for nilpotent groups [Consider S3].(b) If H < C( G) and G/H is nilpotent, then G is nilpotent.Theorem 7.11(i) Every subgroup and every
(a) Let G be the additive group of all polynomials in x with integer coefficients. Show that G is isomorphic to the group Q* of all positive rationaJs (under multiplication).(b) The group Q* is free
Show that every automorphism of S4 is an inner automorphism, and hence S4 ≅ Aut S4.
If G,H,K and J are groups such that G⊕H X Kand G ,..._, H X K and G satisfies both the ACC and DCC on normal subgroups, then K⊕J.
(a) What are the elementary divisors of the group Z2⊕Z9⊕Z35; what are its invariant factors? Do the same for Z26⊕Z42⊕Z49⊕Z200⊕Z1000.(b) Determine up to isomorphism all abelian groups of
Proof that If a group G contains a subgroup (≠ G) of finite index, it contains a normal subgroup (≅G) of finite index.
Every group G of order p2 (p prime) is abelian.
For each prime p the group Z(p∞) satisfies the descending but not the ascending chain condition on subgroups.
Show that the invariant factors of Zm⊕ Zn are (m,n) and [m,n] (the greatest common divisor and the least common multiple) if (m,n) > 1 and mn if (m,n) = 1.
If G is a group, then the ith derived subgroup G(i) is a fully invariant subgroup, whence G(i) is normal.
If |G| = pn, with p > n, p prime, and H is a subgroup of order p, then H is normal in G.
Proof that If H is a subgroup of a finite abelian group G, then G has a subgroup that is isomorphic to G / H.
If N ⊲ and N ∩ G = (e), then N < C(G).
If a normal subgroup N of order p (p prime) is contained in a group G of order pn, then N is in the center of G.

Showing 200 - 300 of 644