Show that the quotient ring Q[x]/(x2 − 3) is isomorphic to a subfield of the real...

Show that the quotient ring Q[x]/(x2 − 3) is isomorphic to a subfield of the real numbers R.

Expert Solution

Colby Messinger answered 1 year ago

3. Show that the Galois group of (x2 − 3)(x2 + 3) over Q is isomorphic...

3. Show that the Galois group of (x2 − 3)(x2 + 3) over Q is isomorphic to Z2 × Z2. 4. Let p(x) be an irreducible polynomial of degree n over a finite field K. Show that its Galois group over K is cyclic of order n.

6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic to the...

6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic to the subfield of C consisting of complex numbers with rational real and imaginary parts.

Show that {a + b√3 | a, b ∈ Q} is a field (a subfield of...

Show that {a + b√3 | a, b ∈ Q} is a field (a subfield of the field R), but {a + b√3 | a, b ∈ Z} is not a field.

If I is an ideal of the ring R, show how to make the quotient ring...

If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

1. If I is an ideal of the ring R, show how to make the quotient...

1. If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

a. Consider d on R, the real line, to be d(x,y) = |x2 – y2|. Show...

a. Consider d on R, the real line, to be d(x,y) = |x2 – y2|. Show that d is NOT a metric on R. b.Consider d on R, the real line, to be d(x,y) = |x3 – y3|. Show that d is a metric on R. 2. Let d on R be d(x,y) = |x-y|. The “usual” distance. Show the interval (-2,7) is an open set. Note: you must show that any point z in the interval has...

discrete structures problems 1.Find a limit to show that x(In(x2))3 is O(x2). Simplify when possible to...

discrete structures problems 1.Find a limit to show that x(In(x2))3 is O(x2). Simplify when possible to avoid doing more work than you have to. You will need to use L'Hôpital's rule at least once. 2.Suppose that f is o(g). What is lim(f(n)/g(n)) as n→ ∞? 3.Suppose that algorithm has run-time proportional to log n and takes 1 millisecond to process an array of size 3,000. How many milliseconds will it take to process an array of size 27,000,000,000 ? Hint:...

Problem 1. Show that the cross product defined on R^3 by [x1 x2 x3] X [y1...

Problem 1. Show that the cross product defined on R^3 by [x1 x2 x3] X [y1 y2 y3] = [(x2y3 − x3y2), (x3y1 − x1y3), (x1y2 − x2y1)] makes R^3 into an algebra. We already know that R^3 forms a vector space, so all that needs to be shown is that the X operator is bilinear. Afterwards, show that the cross product is neither commutative nor associative. A counterexample suffices here. If you want, you can write a program that...

Let R be the ring of all 2 x 2 real matrices. A. Assume that A...

Let R be the ring of all 2 x 2 real matrices. A. Assume that A is an element of R such that AB=BA for all B elements of R. Prove that A is a scalar multiple of the identity matrix. B. Prove that {0} and R are the only two ideals. Hint: Use the Matrices E11, E12, E21, E22.

f(x) = −x2 + 3x and g(x) = x − 3 Find the area of the...

f(x) = −x2 + 3x and g(x) = x − 3 Find the area of the region completely enclosed by the graphs of the given functions f and g in square units.

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