Some vector spaces

The GAP code

vecs:=[[1,0,0],[1,1,1],[0,1,1]];
V:=VectorSpace(Rationals,vecs);
GeneratorsOfVectorSpace(V);
B:=Basis(V);
dim:=Length(B);

constructs the vector space over $\mathbb{Q}$ spanned by the vectors $(1,0,0),(1,1,1),(0,1,1)$, finds a basis, and computes its dimension.

Exercise 2.13.1   Construct the vector space over $\mathbb{Q}$ spanned by the vectors $(1,0,0),(1,1,1),(0,-1,1)$, find a basis, and compute its dimension. Do the same, but with $\mathbb{Q}$ replaced by $GF(3)$.

Exercise 2.13.2   List all the elements of $GF(7)$. Create a multiplication table for $GF(7)$.

The command GF returns a field for any prime power $p^k$. GAP assigns a fixed primitive element to each field $GF(p^k)$: Z(p^k). This is a root of a Conway polynomial, which is the primitive polynomial over $GF(p)$ defining $GF(p^k)$.

Exercise 2.13.3   List all the elements of $GF(25)$.

Exercise 2.13.4   Using GAP, create a multiplication table for $GF(4)$.

The command AlgebraicExtension constructs an extension field.

gap> x:=Indeterminate(Rationals,"x");;
gap> p:=x^4+3*x^2+1;;
gap> e:=AlgebraicExtension(Rationals,p);
<field in characteristic 0>
gap> a:=RootOfDefiningPolynomial(e);
(a)
gap> a^5;
(-1*a-3*a^3)

Exercise 2.13.5   Using GAP, let

gap> x:=Indeterminate(GF(5),"x");;
gap> p:=x^4+x^2+2;;
gap> IsIrreducible(p);
true
gap> F:=AlgebraicExtension(GF(5),p);     
<field of size 625>
gap> a:=RootOfDefiningPolynomial(F);
(a)

Compute $a, a^2, a^3, a^4, a^5, a^6$.