Let K/F be an extension of fields such that [K:F]=p, where p is prime. Show that K = F(a) for every element a of K that is not in F. (Hint: This problem compares 3 fields.)
Prove that every ideal of F[x], where F is any field, is principal
An ideal I of a commutative ring R is said to be principal if it is generated by a single element, that is, if I is of the form {ra|r element of R} for some fixed a element of R. Notice that Corollary 6.6 (below) shows that ideals of F[x] of the form I_F,a are principal. Now prove that every ideal of F[x], where F is any fi ...continues
Definitions - Briefly define a group. Briefly define a ring. Briefly define a field.
Briefly define a group. Briefly define a ring. Briefly define a field.
Please give a real-life example that uses graphing to show data .
Consider the function f:A→A defined by f(x)=x+1 and justify your answers. a) For A=Ν (integers) is f onto? b) For A=R(real number) is f injective? c) For A=Q (rationals) is f onto? d) For A=Z(all integers) is f a bijection?
a) Let f : R→R be given by x→3x-1 and g:R→R be given by x→x+1. Calculate (i) fοg and ii) gοf. b) Prove that fοg and gοf are both invertible and describe their inverses. c) Demonstrate that (fοg)^-1 = g^-1 ο f^-1
find an equation for the line that passes through (1,-5) (-3,5) in (x,Y) coordinates
1a. If X and Y are infinite sets with the same number of elements, show that the following conditions are equivelent for a function f: X-->Y: (i). f is injective (ii). f is bijective (iii) f is surjective 1b. Suppose there are 11 pigeons sitting in some pigeonhole. If there are only 10 pigeonholes prove that there is a h ...continues
Let f: X--->Y and g: Y--->Z be functions. i. If both f and g are injective prove that g o f is injective. ii. If both f and g are surjective prove that g o f is surjective iii. If both f and g are bijective prove that g o f is bijective iv. If g o f is bijection, prove that f is an injection and g is a surjection.
(i) Show that 1000 is congruent to -1 mod 7 (ii) Show that if a= r0(meaning r not) + 1000R1 + 1000^2R2 +..., then a is divisible by 7 if and only if r0-r1+r2-..... is divisible by 7. ( r0- doesn't mean r time zero, I didn't know how to put r not, r one, etc. into my computer with it being a subscript.)
