Question

Prove that the map φ : Q( √ 3) → Q( √ 3) given by a + b √ 3 → a − b √ 3 is an isomorphism of fields.

Answer #1

Prove that the map φ : Q( √ 3) → Q( √ 3) given by a + b √ 3 → a
− b √ 3 is an isomorphism of fields.

Suppose φ : G → G′ is an isomorphism.
(a) Prove that φ(Z(G)) = Z(G′).
(b) Prove that |g| = |φ(g)| for all g ∈ G

Suppose φ:Q→Z is a homomorphism (both groups are under
addition). Prove that φ is the zero map, i.e., φ(x) = 0 for all x ∈
Q.

Is Q(cos φ) = Q(sin φ) for every angle φ?

Let
F and L be fields, and
let φ : F → L be a ring homomorphism.
(a) Prove that either φ is one to one or φ is the trivial
homomorphism.
(b) Prove that if charF= charL, then φ is the trivial
homomorphism

Let φ : A → B be a group homomorphism. Prove that ker φ is a
normal subgroup of A.

(A) Prove that over the field C, that Q(i) and Q(2) are
isomorphic as vector spaces?
(B) Prove that over the field C, that Q(i) and Q(2) are not
isomorphic as fields?

Let
φ:G ——H be a group homomorphism and K=ker(φ). Assume that xK=yK.
Prove that φ(x)=φ(y)

Prove using the definition of truth that for any first-order
formula φ, φ is valid iff ∀x(φ) is valid.

(A) Prove that over the field C, that Q(i) and Q(sqrt(2)) are
isomorphic as vector spaces.
(B) Prove that over the field C, that Q(i) and Q(sqrt(2)) are
not isomorphic as fields

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