15. Show that every element in a finite field can be written as the sum of two squares.

Q: 3. Given the polynomials p1 = t3 – 2t2 + 1, p2 = t2 – 4, p3 = t3 + 2t, p4 = 5t. Does this set of…

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Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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Q: Show that the characteristic of a finite field must be a prime.

A: Characteristic of a finite field must be a prime is as follows : Let 1 denote the multiplicative…

Q: Give a construction of a field with 16 elements.

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Q: 5. For how many different integers bis the polynomial x² + bx + 16 factorable over the integers?

A: From the given data

Q: Recall that {C, +, ·} is the field of all complex numbers. Introduce a total order < on C as…

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Q: Let k be some positive integer. Prove that a number of the form 4k +3 can never be the sum of two…

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Q: Let x, y, z be elements of the real numbers. Use the field axioms and ordered axioms to prove x<y if…

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Q: Can you please identify the splitting field of (x3 + x2 + 1) over Z3.  Write the addition and…

A: Given,Polynomial: (x3+x2+1)We need to identify the splitting field of x3+x2+1 over Z3.To draw the…

Q: Explain why the ring of integers Z under usual addition and multiplication is not a field.

A: Given that the set of integers Z is a ring under addition and multiplication. We know that an…

Q: 1) Name the polynomial a) 3x? + 9x - 1 b) 2x* - 9x? - 1

A: a) 3x2+9x-1The above polynomial has a degree 2⇒It is quadratic polynomial