Recent content by Elzair

Homework Statement Given f\left(x\right) \in \mathbb{R}\left[x\right], if a + bi is a root of f\left(x\right), then a - bi is a root of f\left(x\right) as well. Homework Equations Factor Theorem: if f\left(x\right) is a polynomial over a ring with identity, then a is a root of...

Thanks! I just have one question, though. Why is n-1 the highest exponent? Doesn't F \left( \alpha \right) have degree n?

Homework Statement Let E be an extension of a finite field F, where F has q elements. Let \alpha \epsilon E be algebraic over F of degree n. Prove F \left( \alpha \right) has q^{n} elements. Homework Equations An element \alpha of an extension field E of a field F is algebraic over F if f...

Homework Statement Prove the sequences \left\{ n^{2} \right} _{n \in N} and \left{ -n \right}_{n \in N} do not converge. Homework Equations lim n^2 = a lim -n = b The Attempt at a Solution If n^2 converges, then for all epsilon > 0, there is an N s.t. |n^2 - a| < epsilon for all n > N. If...

Homework Statement Prove the following limit: lim_{n \rightarrow \infty } n^{ 1 / n } = 1 Homework Equations Not sure. The Attempt at a Solution Given any \epsilon > 0, choose N \in \mdseries N s.t. \left| n^{ 1 / n } - 1 \right| < \epsilon for all n > N I am not sure...

Proving a linear system of equations cannot have more than one finite solutions Homework Statement Prove that the number of solutions to a linear system can not be a finite number larger than one. Provide either a general proof or for a system with two equations and two unknowns...

Homework Statement Derive the following equation Homework Equations TdS = C_{V} \left( \frac{\partial T}{\partial P} \right)_{V}dP + C_{P} \left( \frac{\partial T}{\partial V} \right)_{P}dV The Attempt at a Solution dU = \delta Q - \delta W \delta Q = TdS for a closed...

Homework Statement Suppose there exists a relation among three coordinates x, y and z such that f(x,y,z) = 0 Then prove the following relation Homework Equations \left(\frac{\delta x}{\delta y} \right)_{z} = \frac{1}{\left( \frac{\delta y}{\delta x} \right)_{z}} The Attempt...

Thanks! Thanks, I think I got it. I will now post the answer here for others. \frac{dV}{dT} = \frac{\beta}{\kappa} therefore \Delta P = \frac{\beta}{\kappa}\Delta T P.S. Does the Tex function on this forum feature a way to represent the therefore symbol (i.e. a triangle of three dots)?

I am having a little trouble on this problem: Can I treat this as an Isobaric expansion and an Isothermal compression? I think I found the change in Volume like this: \frac{\Delta V}{V_{0}}=\beta\Delta T How do I find the Pressure?