Prove: Every root field over F is the root field of some irreducible polynomial over F....

Prove:

Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)

Expert Solution

Colby Messinger answered 1 year ago

Determine whether the polynomial f(x) is irreducible over the indicated field. (a) f(x) = 4x^ 3...

Determine whether the polynomial f(x) is irreducible over the indicated field. (a) f(x) = 4x^ 3 + 10x^2 + 16x − 11 over Q (b) f(x) = 7x^ 4 − 9x^ 2 + 6 over Q (c) f(x) = x^3 − x + 1 over Q (d) f(x) = x ^4 + 2x ^2 + 2 over Q (e) f(x) = x^ 4 + 2x^ 2 + 2 over C (f) f(x) = x ^4 + 2x^ 2 + 2...

Let f be an irreducible polynomial of degree n over K, and let Σ be the...

Let f be an irreducible polynomial of degree n over K, and let Σ be the splitting field for f over K. Show that [Σ : K] divides n!.

Prove that every polynomial having real coefficients and odd degree has a real root

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A ﬁeld F is said to be perfect if every polynomial over F is separable. Equivalently,...

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Does every polynomial equation have at least one real root? a. Why must every polynomial equation...

Does every polynomial equation have at least one real root? a. Why must every polynomial equation of degree 3 have at least one real root? b. Provide an example of a polynomial of degree 3 with three real roots. How did you find this? c. Provide an example of a polynomial of degree 3 with only one real root. How did you find this?

Please show work and explain Prove that a vector space V over a field F is...

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If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and h(?) ≠ 0. a) Prove that...

If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and h(?) ≠ 0. a) Prove that [?(?)] = [?(?)] if and only if ?(?) ≡ ?(?)(???( (h(?)). b) Prove that congruence classes modulo h(?) are either disjoint or identical.

An element a in a field F is called a primitive nth root of unity if...

An element a in a field F is called a primitive nth root of unity if n is the smallest positive integer such that an=1. For example, i is a primitive 4th root of unity in C, whereas -1 is not a primitive 4th root of unity (even though (-1)4=1). (a) Find all primitive 4th roots of unity in F5 (b) Find all primitive 3rd roots of unity in F7 (c) Find all primitive 6th roots of unity in F7...

Determine whether the polynomial is reducible or irreducible in the given polynomial ring. Justify your answers....

Determine whether the polynomial is reducible or irreducible in the given polynomial ring. Justify your answers. (c) x^4 + 1 in Z5[x] (e) 2x^3 − 5x^2 + 6x − 2 in Z[x] (f) x^4 + 4x^3 + 6x^2 + 2x + 1 in Z[x]. Hint: Substitute x − 1 for x.

Use induction to prove Let f(x) be a polynomial of degree n in Pn(R). Prove that...

Use induction to prove Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the nth derivative of f(x).

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