Give an extension field of $\mathbb{Z}_3$ of degree 3? I have an irreducible polynomial in $\mathbb{Z} $ That irreduible polynomial is:
$(1)$ $x^3 + 2x + 1$ 
I know that this polynomial creates a maximal ideal and that I can create an extension field from the quotient ring. 
What I don't know how to do is be sure of my extension's degree. What are some ways I can verify my new extension fields degree? I know if I just considered $(1)$ It can be looked at like a vector space, and I would think it is degree 3 because I know it is irreducible in the field. But again, not too sure
EDIT:
I'm reading in my textbook the following: 
Suppose that E/F is a field extension. Then E may be considered as a vector space over F, where F is a field of scalars. 
Our E is all polynomials in $\mathbb{Z}_3[x]$ then. 
Can I say that this automatically has degree 3 because I have $x^3 + 2x + 1$ in this field E? And I know it is irreducible and it is looked at like a vector space. 

Yes, $f(x)=x^3+2x+1$ is irreducible over the field $\mathbb{F}_3$ with $3$ elements.
Hence the result follows from the following proposition, with $n=p=3$.
Proposition: For a prime p and a monic irreducible $f(x)$ in $\mathbb{F}_p[x]$ of degree $n$, the ring $\mathbb{F}_p[x]/(f(x))$ is a field of order $p^n$. In particular, the degree is $[\mathbb{F}_p[x]/(f(x)):\mathbb{F}_p]=n$.
