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…

A:

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

A:

Q: Give and write 5 example each of polynomials with degree 3, 4, 5, and 2.

A: If a polynomial is degree n then highest degree should be n with non zero coefficient

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: Need correct answer, Show that x2 + 3 and x2 + x + 1 over Q have same splitting field.

A: Solution:-

Q: . Show that x2 + 3 and x2 + x + 1 over Q have same splitting field.

A:

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

A:

Q: Show that the ring R={x + y√√2; x,y is under the set of real numbers} is a field

A:

Q: Which of the following polynomials is reducible over Q? * x^4+x+1 O 3x^2+x+2 3x^3-6x^2+x-2 None of…

A:

Q: Determine which of the polynomials below is (are) irreducibleover Q.

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: 3. A polynomial containing exactly two terms is a(n) .......................

A: We know that, A polynomial containing exactly one term is a monomial. A polynomial containing…

Q: In real analysis,Prove that if A is countable, then m(A) =0.

A:

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

A: From the given data

Q: 89. Prove that the product of two cubic polynomials is apolynomial of degree six.

A: Suppose two general cubic polynomials are px=a3x3+a2x2+a1x+a0andqx=b3x3+b2x2+b1x+b0where a3≠0, b3≠0

Q: 5. if we have two polynomials pl=3x5 + 8x* – 2x2 + 4x + 1 p2= 2x6 + 3x* – x² + 4x Find:- Your answer…

A:

Q: ) Use the distributive law to determine which answer represents the given expression as a polynomial…

A: Distributive law a(b + c) = ab + ac

Q: Determine whether the expression is a polynomial or not. A) Polynomial B) NOT a polynomial 7. 2x -x…

A:

Q: Show that the polynomial 1+x+x2 +...+xP-1 , where p is a prime number, is irreducible over the field…

A:

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

A:

Q: 6.Do the polynomials P1 = 1 – x + 2x², p2 = 3 + x, p3 = 5 – x + 4x², p4 = -2 – 2x + 2x² span P2 ?…

A:

Q: 21. Suppose A is a field and suppose also that A closed under countable disjoint unions. Show A

A: This is a question of measure theory.

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

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

A:

Q: Determine the degree of the splitting field of x4 3 over Q. Justify your - answer.

A:

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

A:

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: Can a Irreducible polynomial be a field with a degree that is larger or the same as 1?

A: Can a Irreducible polynomial be a field with a degree that is larger or the same as 1?

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: Show that the set QR/2) = {a + b{2 +c4:a,b,ce Q} with the usual actions of + and*and with the…

A: Given set is, Q23=a+b23+c43 Where a,b,c∈Q For sum of two elements,…

Q: 1. How do you know that the given polynomial is divisible or not? 2. How will you prove that the…

A: Given: A polynomial f(x) To check: 1) Whether the polynomial is divisible or not.

Q: 1. Compute the resultant of x – 3x* – 2r + 3x + 7x + 6 and x* +² + 1. Do polynomials have a common…

A: Given: To calculate the resultant of .

Q: is Zs is a field ?if yes show it if not give a reason

A:

Q: Find ntegers G and t such that gcd (963,652)3967+t-652,

A:

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