Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D_2p of a regular p-gon. Prove that f (x) has either all real roots or precisely one real root.
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group...
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...
5. Let F be a field, and let p(x) ∈ F [x] be a separable, irreducible polynomial of degree 3. Let K be the splitting field of p(x), and denote the roots of p(x) in K by α1, α2, α3. a) (10’) If char(F ) does not equal 2, 3, prove that K = F (α1 − α2).
Suppose that p is prime. For a regular p-gon to be Euclidean constructible, then the roots of r- 1 must be constructible. The roots of zp-4z?-2+ . . . +z+1 together with 1-1 form a regular p-gon. They would need to be constructible. Since z-1 +···+ z + 1 is irreducible, that means the degree of a root of this is p-1. Using this prove that p 22 + 1 for some nonnegative integer n
Suppose that p is prime....
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...
Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hint: Assume it is reducible. If one of the factors has degree m, look at the coefficient of x m−1 and get a contradiction.]
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()...
Computing another Galois Group a) Let? = eri/6 be a primitive 12th root of unity. Prove that is a zero of the polynomial t4 - t+ 1, and that the other zeros are 55,57,511. b) Prove that t4 – +2 + 1 is irreducible over Q and is the minimal polynomial of over Q.
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...
17. The real number a = cos 20° is a root of the irreducible polynomial f(x) = 4x? – 3x 3x = in Q[x]. Let E = Q[cos 20°). Show that f(x) splits in E.
76.Let p be an odd prime. Prove that if Ord, (a) = his even, then a/2 = -1 mod p. 77.let p be an odd prime. Prove that if Ord, (a) = 3, then 1+ a + a? = 0 mod p and Ord,(1 + a) = 6. 78.Show that 3 is a primitive root modulo 17. How many primitive roots does 17 have? Find them.