10. Suppose that f(x) E Z[x] and f(x) is irreducible over Z, where p is a...
Part D,E,F,G 10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either all real roots precisely one real root or 2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either...
Example 4.2.4 shows f=x^n+px+p with p prime implies that f is irreducible over Q by Eisenstein criterion Exercise 1. Lemma 4.4.2 shows that a finite extension is algebraic. Here we will give an example to show that the converse is false. The field of algebraic numbersis by definition algebraic over Q. You will show that :Ql oo as follows. (a) Given n 22 in Z, use Example 4.2.4 from Section 4.2 to show that @ has a subficld such that...
Rings and fields- Abstract Algebra 2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g() e Fx be any polynomial. Show that every irreducible factor of f(g()) E Flx] has degree divisible by n (b) (4 points) Prove that Q(2) is not a subfield of any cyclotomic field over Q. 2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g()...
Prove or Disprove: Let p E P(F) and suppose that deg p > 1 and p is irreducible. Then p(a)メ0 for all a E F.
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
Exercise 2 (pts 5). Let g() E Z[2]. Prove that g(x) is irreducible over Zx if and only if g() is irreducible as polynomial in Q[o].
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...
A finite field is any finite extension of Fp := Z/pZ. The characteristic of a field F is the generator of the kernel of the map ι : Z → F, ι(1) = 1. (a) Prove that there exist finite fields of order pnfor any prime p. We denote such a field Fpn. (b) Prove that Fpn has characteristic p. (c) Prove that the Frobenius map φ(a) = ap is an automorphism of Fpn . (d) If f(x) ∈ Fpn...
4. (a) Prove that if f(x) E Q[x] is irreducible in R[x], then it is irreducible in Q[x]. Is the converse of this statement true? Explain why or why not. (b) Prove that if f(x) E Q[x] is reducible in Q[x], then it is reducible in R[x]. Is the converse of this statement true? Explain why or why not.