Prove that the polynomial 9x^4 + 4x^2 − 3x + 7 is irreducible in Q[x].
Prove that the polynomial 9x^4 + 4x^2 − 3x + 7 is irreducible in Q[x].
Prove that x4 + 3x + 4x² + 8x + 11 is irreducible in Q[x] . Make sure to completely justify all your claims.
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.
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.
2. Consider the polynomial p = x3 + x +4 € Z5 [2]. Let q = 3x +2 € Z5 [2]. (a) Is p reducible or irreducible? Prove your claim. (b) Are there any degree 2 polynomials in [g],? Explain. (c) List all degree 3 polynomials in [g]p. (d) (ungraded for thought) How many degree 4 polynomials are in (q),? Degree 5?
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()...
Write the polynomial f(x) as a product of irreducible polynomials in the given ring. Explain in each case how you know the factors are irreducible. 1) f(x) -x* + 2x2 +2x 2 in Z3[x]. 2) f(x)4 + 2x3 + 2x2 +x + 1 in Z3[x]. 3) f(x) 2x3-x2 + 3x + 2 in Q[x] 4) f(x) = 5x4-21x2 + 6x-12 in Q[x)
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...
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...
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].
(8) Show that each polynomial is irreducible in Q[x]. (a) 3x3 + 5x2 + x + 2 (b) 23 + 9x2 + x + 6