Let ?∈ℕ, and assume √? is irrational. Show that ℚ(√?)={?+?√?∶?,?∈ℚ} is a field (show that there...

Let ?∈ℕ, and assume √? is irrational. Show that ℚ(√?)={?+?√?∶?,?∈ℚ} is a field (show that there is multiplicative commutativity and multiplicative inverse). What would change if ℚ was replaced with ℝ.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let a be a positive element in an ordered field. Show that if n is an...

Let a be a positive element in an ordered field. Show that if n is an odd number, a has at most one nth root; if n is an even number, a has at most two nth roots.

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 vector field. Find the flux of F through the given surface. Assume...

Let F be a vector field. Find the flux of F through the given surface. Assume the surface S is oriented upward. F = eyi + exj + 24yk; S that portion of the plane x + y + z = 6 in the first octant.

4. Use a proof by contradiction to show that the square root of 3 is irrational....

4. Use a proof by contradiction to show that the square root of 3 is irrational. You may use the following fact: For any integer k, if k2 is a multiple of 3, then k is a multiple of 3. Hint: The proof is very similar to the proof that √2 is irrational. 5. Use a direct proof to show that the product of a rational number and an integer must be a rational number. 6. Use a proof by...

Discrete Structures Use a proof by contraposition to show if x3 + 3x is an irrational...

Discrete Structures Use a proof by contraposition to show if x3 + 3x is an irrational number then so is x, for any real number x.

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)...

Show that for any square-free integer n > 1, √ n is an irrational number

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.

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.

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...

Subjects