Algebra II – Chapter 6 Day #5
... I can use the Rational Root Theorem to solve equations. I can use the Conjugate Root Theorem to solve equations. I can use the Descartes’ Rule of Signs to determine the number of roots of a polynomial equation. I can use synthetic division to divide two polynomials. We want to first look at ...
... I can use the Rational Root Theorem to solve equations. I can use the Conjugate Root Theorem to solve equations. I can use the Descartes’ Rule of Signs to determine the number of roots of a polynomial equation. I can use synthetic division to divide two polynomials. We want to first look at ...
Introduction to Polynomials and Polynomial Functions
... An algebraic term is a number or a product of a number and a variable (or variables) raised to a positive power. Examples: 7x or -11xy2 or 192 or z A constant term contains only a number A variable term contains at least one variable and has a numeric part and a variable part A polynomial expression ...
... An algebraic term is a number or a product of a number and a variable (or variables) raised to a positive power. Examples: 7x or -11xy2 or 192 or z A constant term contains only a number A variable term contains at least one variable and has a numeric part and a variable part A polynomial expression ...
INTRODUCTION TO ALGEBRA II MIDTERM 1 SOLUTIONS Do as
... Solution: The powers of 3 modulo 19 are 3, 9, 8, 5, 15, 10, 11, 14, 4, . . . . No need to go on, since we already see that the order of 3 in the multiplicative group Z∗19 (which has 18 elements) is at lest 10. Since by Lagrange’s theorem this order divides 18, it is 18, and 3 is a generator. b.– How ...
... Solution: The powers of 3 modulo 19 are 3, 9, 8, 5, 15, 10, 11, 14, 4, . . . . No need to go on, since we already see that the order of 3 in the multiplicative group Z∗19 (which has 18 elements) is at lest 10. Since by Lagrange’s theorem this order divides 18, it is 18, and 3 is a generator. b.– How ...
FINAL EXAM
... (8) (12 points) Let p(x) = x2 + bx + c be an irreducible, monic, quadratic polynomial in Q[x], and let K = Q[x]/(p(x)). (a) Show that K contains all the roots of p(x). ...
... (8) (12 points) Let p(x) = x2 + bx + c be an irreducible, monic, quadratic polynomial in Q[x], and let K = Q[x]/(p(x)). (a) Show that K contains all the roots of p(x). ...
FINAL EXAM
... Deduce that, in the notation of (a), if OK has d generators over Z, then [K : Q] ≤ 2d . (3) Let ζ be a 151-th root of 1, L = Q(ζ). Show that the cyclotomic field L contains a unique subfield K of degree 10 over Q. (Check, but don’t bother writing down, that 151 is a prime number.) Show that ...
... Deduce that, in the notation of (a), if OK has d generators over Z, then [K : Q] ≤ 2d . (3) Let ζ be a 151-th root of 1, L = Q(ζ). Show that the cyclotomic field L contains a unique subfield K of degree 10 over Q. (Check, but don’t bother writing down, that 151 is a prime number.) Show that ...