Let F be a field, and recall the notion of the characteristic of a ring; the...

Let F be a field, and recall the notion of the characteristic of a ring; the characteristic of a field is either 0 or a prime integer.

Show that F has characteristic 0 if and only if it contains a copy of rationals and then F has characteristic p if and only if it contains a copy of the field Z/pZ.

Show that (in both cases) this determines the smallest subfield of F.

Solutions

Expert Solution

Colby Messinger answered 11 months ago

Next > < Previous

Related Solutions

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...

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.

2 Let F be a field and let R = F[x, y] be the ring of...

2 Let F be a field and let R = F[x, y] be the ring of polynomials in two variables with coefficients in F. (a) Prove that ev(0,0) : F[x, y] → F p(x, y) → p(0, 0) is a surjective ring homomorphism. (b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x, y) + ys(x, y) | r,s ∈ F[x, y]} (c) Use the first isomorphism theorem to prove that (x, y) ⊆ F[x, y]...

Let A be a commutative ring and F a field. Show that A is an algebra...

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

Let F be a field and R = Mn(F) the ring of n×n matrices with entires...

Let F be a field and R = Mn(F) the ring of n×n matrices with entires in F. Prove that R has no two sided ideals except (0) and (1).

Let D be a division ring, and let M be a right D-module. Recall that a...

Let D be a division ring, and let M be a right D-module. Recall that a subset S ⊂ M is linearly independent (with respect to D) if for any finite subset T ⊂ S, and elements at ∈ D for t ∈ T, if sum of tat = 0, then all the at = 0. (a) If S ⊂ M is linearly independent, show that there exists a maximal linearly independent subset U of M that contains S, and...

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)...

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring...

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring of the functions from X to R. Show directly the associativity of the multiplication of F(X, R). Assume that R is unital and commutative. show that F(X, R) is also unital and commutative.

let E be a finite extension of a ﬁeld F of prime characteristic p, and let...

let E be a finite extension of a ﬁeld F of prime characteristic p, and let K = F(Ep) be the subﬁeld of E obtained from F by adjoining the pth powers of all elements of E. Show that F(Ep) consists of all ﬁnite linear combinations of elements in Ep with coeﬃcients in F.

Let R be a UFD and let F be a field of fractions for R. If...

Let R be a UFD and let F be a field of fractions for R. If f(α) = 0, where f ∈ R [x] is monic and α ∈ F, show that α ∈ R NOTE: A corollary is the fact that m ∈ Z and m is not an nth power in Z, then n√m is irrational.

Subjects