This page contains the PS assignments that students had to solve.
(§41, Ex. 1) Let V be a vector space over a field
K and let W be a subgroup of the additive group (V, +). For all
a in K and for all v + W in the factor
group V/W, we write a o (v + W) = a v + W. Prove that (a,
v + W) —> a o (v + W) is a well-defined
mapping from K × V/W to V/W if and only if W is a subspace
of V.
(§41, Ex. 2) Let f : V —> V1 be a K-vector space homomorphism, let W be a subspace of V such that W ≤ Ker f and let n : V to V/W be the associated natural homomorphism. Show that there is a K-vector space homomorphism y : V/W —> V1 such that ny = f and Ker y = (Ker f)/W. What happens if we drop the condition W ≤ Ker f?
(§42 Ex. 1) Let V be a vector space over a field K and let W be a subspace of V. Show that there is a subspace U of V such that V = W + U and W intersection U = {0}.
(§42, Ex. 5) Find all R-linear mappings from R4 onto R5.
(§48, Ex. 2) Let p be prime. Is Zp2 an extension of Zp? Is Zp3 an extension of Zp2?
(§48, Ex. 12) Let K be a field of characteristic p ≠ 0. Prove that f : K ; K, a —> ap is a field homomorphism.
(§48 Ex. 8) Show that Q(√2, i) := {a + b√2 +c i + d√2 i : a, b, c, d in Q} is an extension field of both Q(i) and Q(√2). Find |Q(√2, i) : Q| by two different methods.
(§48, Ex. 6) Find three nonisomorphic fields
of characteristic p ≠ 0.
(§48, Ex. 10) Let K be a field and e the identity element of K. Show that char K = 0 or p according as the subring of K generated by e is isomorphic to Z or to Zp.
(§49, Ex. 3) Let E/K be a field extension and S a subset of E. Prove that K(S) = K if and only if S is a subset of K.
(§49, Ex. 9) Prove that every finitely dimensional extension is finitely generated. Find a finitely generated field extension that is not finite dimensional.
(§49, Ex. 6) Show that √2 + i, √2 + √3, √2 + √3 + i are algebraic over Q by exhibiting polynomials in Q[x] having these numbers among their roots.
(§50 Ex. 2) Find the minimal polynomial of the
following numbers over the indicated fields.
√2 over Q;
√3 - √2 over Q, over Q(√2), over
Q(√3).
(§50 Ex. 2) Let E/K be a field extension and let D be an integral domain such that K is a subset of D and D is a subset of E. Prove that, if E is algebraic over K, then D is a field.
(§51 Ex. 4) Find a field K of 9 elements and show that K× is cyclic.
(§49 Ex. 10) Let E/K be a field etension, x, y indeterminates over K and let a, b be elements of E. Prove or disprove: if both a and b are transcendental over K, then K(a, b) is isomorphic to K(x, y) (cf. Theorem 49.10).
Prove that F5× and F5(√2)× are cyclic.
Let K = F5(√2). Find a field in which the polynomial x2-√2 (in K[x]) has a root; and find a field in which the polynomial x2-(2 + 2√2) (in K[x]) has a root.
(§49 Ex. 7) Let K be a field. Prove that every element in K(x) that doesn't belong to K is transcendental over K.
(§50 Ex. 4) Let E/K be a field extension and let a, b be elements of E, both algebraic over K. If the degree of a over K is m and the degree of b over K is n, and if m, n are relatively prime, show that K(a, b) is algebraic over K and |K(a, b) : K| = mn.
(§52 Ex. 2) Let E and K be finite fields, K contained in E, such that |E : K| = 5. Let a be an element of K. If there is no b in K such that b2 = a, show that there is no b in E such that b2 = a.
(§52 Ex. 3) Let E and K be finite fields, K contained in E, and let n := |E : K|. Let a be an element of K. If n is odd and if there is no b in K such that b2 = a, prove that there is no b in E such that b2 = a.
(§52 Ex. 3) Let E and K be finite fields, K contained in E, and let n := |E : K|. Let a be an element of K. If n is even and if there is no b in K such that b2 = a, prove that there is some b in E such that b2 = a.
(§52 Ex. 4) Let p and q be distinct prime numbers. Find the number of monic irreducible polynomials of degree q in Fp[x].
Özge Sevil (§53 Ex. 1) Construct a splitting of x3 - 3, x4 - x2 + 1 over Q.
Let E/K be the field extension indicated below. Find the Galois group AutKE and all its subgroups, and describe the Galois correspondence between the subgroups of AutKE and the intermediate fields of E/K.
(§54 Ex. 3) Let E/K be a field extension and G := AutKE. Let L, M be intermediate fields of E/K, and let H, J be subgroups of G. Prove that <H union J>' = H' intersection J' and (LM)' = L' intersection M'.
(§55 Ex. 8) Find a primitive element for the extension Q(√2, √3) of Q.
(§55 Ex. 8) Find a primitive element for the extension Q(√2, √3, √5) of Q.
(§55 Ex. 8) Find a primitive element for the extension Q(√2, i) of Q.
Find a field extension that is not separable.
(§55 Ex. 15) Let E/K be a finite dimensional field extension. Prove that if E is perfect, then K is perfect.
(§55 Ex. 15) Prove that every finite dimensional extension of a perfect field is perfect.