Let p be a prime congruent to -1 mod 4. Show that X^2 + 1 is irreducible in Z_p[X], and hence K = Z_p[X] / (X^2 + 1) is the field of order p^2. Note that K has a multiplication similar to that of the complex numbers.Proof:
Since p = -1 (mod 4), then -1 is not a quadratic residue of p. It means that the equation x^2 = -1 (mod p) has no solution.
It also means that x^2 + 1 = 0 (mod p) has no solution. Thus x^2 + 1 is irreducible in Z_p[x].
Hence the corresponding field Z_p[x] / (x^2 + 1) has p^2 elements, or has the order p^2.
Each element has the form ax + b, where a, b can be 0, 1, 2, …, p-1.
Done.
Why Choose Us
- 100% non-plagiarized Papers
- 24/7 /365 Service Available
- Affordable Prices
- Any Paper, Urgency, and Subject
- Will complete your papers in 6 hours
- On-time Delivery
- Money-back and Privacy guarantees
- Unlimited Amendments upon request
- Satisfaction guarantee
How it Works
- Click on the “Place Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
- Fill in your paper’s requirements in the "PAPER DETAILS" section.
- Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
- Click “CREATE ACCOUNT & SIGN IN” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page.
- From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.