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For an extension L/K, let a € L be an algebraic element...
Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of ...
Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of α over F is n Then every β E E can be expressed uniquely in the form β-bo-b10 + +b-1a-1 for some bi F. (a) Show every element can be written as f (a) for some polynomial f(x) E F (b) Let m(x) be the minimal polynomial of α over F. Write m(x) r" +an-11n-1+--+ n_1α α0. Use this...
Let k C F = kla) be a simple algebraic extension. Prove that F is normal...
Let k C F = kla) be a simple algebraic extension. Prove that F is normal over k if and only if for every algebraic extension FC K and every o E Autk(K),(F) = F.
An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension,...
An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension, and 2) every nonconstant polynomial in K[x] has a root in K. If K is an algebraic closure of F, prove that every polynomial p(x) ∈ F[x] splits in K[x].
9. Let E be an extension field of a field F. (1) What does it mean...
9. Let E be an extension field of a field F. (1) What does it mean for an element z EE being algebraic over F? (2) What does it mean for an element z E E being transcendental over F? (3) Can you find any element r e C such that r is transcendental over Q? (4) Can you prove that if a E E is algebraic over F then (F(a): F] is finite? (5) Can you prove that if...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
(a) (b) True or False: If K is a field and u, v are algebraic elements of degrees m, n respectively in some extension field of K, then (K(u,v Explain. If you are not sure, explain what you are...
(a)
(b)
True or False: If K is a field and u, v are algebraic elements of degrees m, n respectively in some extension field of K, then (K(u,v Explain. If you are not sure, explain what you are not sure about. Let E be an extension field of K, and let S (u_1,..u_n) be a set of elements of E. Explain what it means for S to be a *basis* for E as a vector space over K. *...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...
7.3.1 Let U be a finite-dimensional vector space over a field F and T є L(U). Assume that λ0 E F ...
7.3.1 Let U be a finite-dimensional vector space over a field F and T є L(U). Assume that λ0 E F is an eigenvalue of T and consider the eigenspace Eo N(T-/) associated with o. Let. uk] be a basis of Evo and extend it to obtain a basis of U, say B = {"l, . . . , uk, ul, . . . ,叨. Show that, using the matrix representation of T with respect to the basis B, the...
ring over Q in countably many variables. Let I be the ideal of R generated by all polynomials -Pi where p; is the ith prime. Let RnQ1,2, 3, n] be the polyno- mial ring over Q in n variables. Let...
ring over Q in countably many variables. Let I be the ideal of R generated by all polynomials -Pi where p; is the ith prime. Let RnQ1,2, 3, n] be the polyno- mial ring over Q in n variables. Let In be the ideal of Rn generated by all ] be the polynomial rin 9. Let R = QlX1,22.Zg, 2 polynomials -pi, where pi is the ith prime, for i1,.,n. . Show that each Rn/In is a field, and that...
Let k C F = kla) be a simple algebraic extension. Prove that F is normal over k if and only if for every algebraic extension FC K and every o E Autk(K),(F) = F.
9. Let E be an extension field of a field F. (1) What does it mean for an element z EE being algebraic over F? (2) What does it mean for an element z E E being transcendental over F? (3) Can you find any element r e C such that r is transcendental over Q? (4) Can you prove that if a E E is algebraic over F then (F(a): F] is finite? (5) Can you prove that if...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
(a)
(b)
True or False: If K is a field and u, v are algebraic elements of degrees m, n respectively in some extension field of K, then (K(u,v Explain. If you are not sure, explain what you are not sure about. Let E be an extension field of K, and let S (u_1,..u_n) be a set of elements of E. Explain what it means for S to be a *basis* for E as a vector space over K. *...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...
7.3.1 Let U be a finite-dimensional vector space over a field F and T є L(U). Assume that λ0 E F is an eigenvalue of T and consider the eigenspace Eo N(T-/) associated with o. Let. uk] be a basis of Evo and extend it to obtain a basis of U, say B = {"l, . . . , uk, ul, . . . ,叨. Show that, using the matrix representation of T with respect to the basis B, the...
ring over Q in countably many variables. Let I be the ideal of R generated by all polynomials -Pi where p; is the ith prime. Let RnQ1,2, 3, n] be the polyno- mial ring over Q in n variables. Let In be the ideal of Rn generated by all ] be the polynomial rin 9. Let R = QlX1,22.Zg, 2 polynomials -pi, where pi is the ith prime, for i1,.,n. . Show that each Rn/In is a field, and that...
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