Prove the theorem in the lecture:Euclidean Domains and UFD's Let F be a field, and let...

Prove the theorem in the lecture:Euclidean Domains and UFD's

Let F be a field, and let p(x) in F[x]. Prove that (p(x)) is a maximal ideal in F[x] if and only if p(x) is irreducible over F.

Expert Solution

Denote I=<p(x)>.Then we have I is principal, I = <p(x)> for some p(x) ∈ F[x].
1. (⇐) Assume p(x) is irreducible and let I1 be an ideal of F[x] containing
I. Then I1 = <f(x)> for some f(x) ∈ F[x]. Since p(x) ∈ I1 we have p(x) =
f(x)q(x) for some q(x) ∈ F[x]. Since p(x) is irreducible this means that either
f(x) has degree zero (i.e. is a non-zero element of F) or q(x) has degree zero.
If f(x) has degree zero then f(x) is a unit in F[x] and I1 = F[x]. If q(x) has
degree zero then p(x) = af(x) for some nonzero a ∈ F, and f(x) = a^(-1)p(x);
then f(x) ∈ I and I1 = I. Thus either I1 = I or I1 = F[x], so I is a maximal
ideal of F[x].
(⇒): Suppose I = <p(x)> is a maximal ideal of F[x]. Then p(x) =/= 0. If
p(x) = g(x)h(x) is a proper factorization of p(x) then g(x) and h(x) both
have degree at least 1 and <g(x)> and <h(x)> are proper ideals of F[x] properly
containing I. This contradicts the maximality of I, so we conclude that p(x)
is irreducible. This completes the proof.

Colby Messinger answered 2 years ago

Prove the following: (a) Let A be a ring and B be a field. Let f...

Prove the following: (a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal. (b) If A/J is a field, then J is a maximal ideal.

Theorem: Let K/F be a field extension and let a ∈ K be algebraic over F....

Theorem: Let K/F be a field extension and let a ∈ K be algebraic over F. If deg(mF,a(x)) = n, then 1. F[a] = F(a). 2. [F(a) : F] = n, and 3. {1, a, a2 , ..., an−1} is a basis for F(a).

Let F be a finite field. Prove that the multiplicative group F*,x) is cyclic.

A theorem for you to prove: Let B be a local ring containing a perfect field...

A theorem for you to prove: Let B be a local ring containing a perfect field k that is isomorphic to its residue field B/m, and such that B is a localization of a finitely generated k-algebra. Then the module of relative differential forms M_B/k is a free B-module of rank equal to the dimension of B if and only if B is a regular local ring.

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths. (b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F...

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F = 0_F . Show further that the smallest positive integer with this property is a prime number.

The goal of this exercise is to prove the following theorem in several steps. Theorem: Let...

The goal of this exercise is to prove the following theorem in several steps. Theorem: Let ? and ? be natural numbers. Then, there exist unique integers ? and ? such that ? = ?? + ? and 0 ≤ ? < ?. Recall: that ? is called the quotient and ? the remainder of the division of ? by ?. (a) Let ?, ? ∈ Z with 0 ≤ ? < ?. Prove that ? divides ? if and...

Let F be a field and let φ : F → F be a ring isomorphism....

Let F be a field and let φ : F → F be a ring isomorphism. Define Fix φ to be Fix φ = {a ∈ F | φ(a) = a}. That is, Fix φ is the set of all elements of F that are fixed under φ. Prove that Fix φ is a field. (b) Define φ : C → C by φ(a + bi) = a − bi. Take for granted that φ is a ring isomorphism (we...

Let D and D'be integral domains. Let c = charD and c'= charD' (a) Prove that...

Let D and D'be integral domains. Let c = charD and c'= charD' (a) Prove that the direct product D ×D'has unity. (b) Let a ∈D and b∈D'. Prove that (a, b) is a unit in D ×D'⇐⇒ a is a unit in D, b is a unit in D'. (c) Prove that D×D'is never an integral domain. (d) Prove that if c, c'> 0, then char(D ×D') = lcm(c, c') (e) Prove that if c = 0, then char(D...

Let (F, <) be an ordered field, let S be a nonempty subset of F, let...

Let (F, <) be an ordered field, let S be a nonempty subset of F, let c ∈ F, and for purposes of this problem let cS = {cx | x ∈ S}. (Do not use this notation outside this problem without defining what you mean by the notation.) Assume that c > 0. (i) Show that an element b ∈ F is an upper bound for S if and only if cb is an upper bound for cS. (ii)...

Question