Construct the affine plane AP(Z3). Determine its parallel classes and the corresponding Latin squares for the classes of finite nonzero slope.

How many polynomials are there of degree ≤ 2 in

$$ℤ_5[X]?$$

(Include 0.)

How many polynomials are there of degree ≤ 3 in

$$ℤ_2[x]?$$

(Include 0.)

Find the following percentiles for the standard normal variable z: a. 90th percentile b. 95th percentile c. 98th percentile d. 99th percentile

Simplify each expression. $\dfrac{b^2}{b^3}$

Find all solutions of the equation x² + 2x + 2 = 0 in

$$ℤ_6.$$

Make a table of values of φ(n) for n ≤ 30.

Find all solutions of the equation x³ - 2x² - 3x = 0 in

$$ℤ_{12}.$$

Find all solutions of x² + 2x + 4 = 0 in

$$ℤ_6.$$

Solve the equation 3x = 2 in the field

$$ℤ_7$$

; in the field

$$ℤ_{23}.$$

Find an example of a $(7,7,4,4, \lambda)-$design.

Let p be a prime. (a) How many monic quadratic (degree 2) polynomials x2 + bx + c in Zp[x] can we factor into linear factors in Zp[x]? (For example, if p = 5, then the polynomial x2 + 2x + 2 in Z5[x] would be one of the quadratic polynomials for which we should account, under these conditions.) (b) How many quadratic polynomials ax2 + bx + c in Zp[x] can we factor into linear factors in Zp[x]? (c) How many monic quadratic polynomials x2 + bx + c in Zp[x] are irreducible over Zp? (d) How many quadratic polynomials ax2 + bx + c in Zp[x] are irreducible over Zp?

a) Let $\mathrm{Q}[\sqrt{2}]=(\mathrm{a}+\mathrm{b} \sqrt{2} | \mathrm{a}, \mathrm{b} \in \mathrm{Q}).$ Prove that $(Q[\sqrt{2}],+, \cdot)$ is a subring of the field $(R,+, \cdot).$ (Here the binary operations in R and $\mathrm{Q}[\sqrt{2}]$ are those of ordinary addition and multiplication of real numbers.) b) Prove that $\mathrm{Q}[\sqrt{2}]$ is a field and that Q[x]/(x2 – 2) is isomorphic to $\mathrm{Q}[\sqrt{2}].$

Construct a finite field of 27 elements.