We review Galois theory, starting with the basic definitions.  A finite extension $L$ of a field $K$ is called {\sl Galois} over $K$ if the order of its Galois group Aut$_K(L)$ (the group fo $K$-automorphisms of $L$) matches the degree $[L:K]$ of the extension.  A weaker property is
{\sl separability}:  $L$ is separable over $K$ if there is an extension $M$  of $K$ with $[L:K]$ many distinct $K$-embeddings from $L$ into $M$ (there can never be more than this number).  This holds if and only if $L$ lies in the splitting field of some polynomial $q$ with no multiple roots over $K$, so it is automatic in characteristic 0 (or for finite fields), but not in general.  If $L$ is Galois over $K$ then there is a 1-1 inclusion-reversing correspondence between fields lying between $K$ and $L$ on the one hand and subgroups of the Galois group $G$ on the other, sending a subgroup $H$ to the fixed field $L^H$ consisting of thel elements of $L$ fixed by all $h\in H$.  Then $L$ is always Galois over $L^H$, with Galois group $H$, but $L^H$ is Galois over $K$ if and only if $H$ is normal in $G$; in this case the Galois group is the quotient group $G/H$.  Any $K$-automorphism of an intermediate field $K'$ between $K$ and $L$ extends to a $K$-automorphism of $L$, if $L$ is Galois over $K$.

You should be as familiar as you can with the basic examples, the most important of which is the splitting field $C_n \ \Bbb Q[e^{2\pi i/n}]$ of $x^n -1$ over $\Bbb Q$.  This is Galois with Galois group $\Bbb Z_n^*$, the {\sl multiplicative} group of units in $\Bbb Z_n$ (so that in particular it is cyclic of order $p-1$ if $n=p$ is prime).  The degree of the extension is $\phi(n)$, the Euler \phi$-function evaluated at $n$, which counts the number of positive integers less than $n$ and relatively prime to it; the minimal polynomial of $e^{2\pi i/n}$ over $\Bbb Q$ is $\Phi_n$, the $n$th cyclotomic polynomial  (which is monic with integer coefficients).  (In general the splitting field of $x^n - 1$ over any field $K$ of characteristic not dividing $n$ is a subgroup of $\Bbb Z_n^*$.)  Then you should look into the other closely related examples that we have seen in class:  the Galois groups of (the splitting fields of)
$x^3 - 2$ and $x^4 - 2$ over $\Bbb Q$ are both dihedral (of respetive orders 6 and 8), but the pattern breaks at the next step, the Galois group of $x^5 - 2$ over $\Bbb Q$ having order 20 and that of
$x^8 - 2$ having order 16, but not being dihedral.  The Galois group of $x^n - a$ over any field $K$ having a primitive $n$th root of 1 (and of characteristic not dividing $n$) is a subgroup of the cyclic group $\Bbb Z_n$, and so in particular is abelian (even cyclic).  This simple observation plays a crucial role in the proof of Galois's famous criterion for solvability of a polynomial by radicals:  {\sl a polynomial $q$ over a field $K$ of characteristic 0 is solvable by radicals if and only if its Galois group is solvable}.  The converse of the fact we just quoted that was used to prove this is of equal importance:  {\sl a Galois extension $L$ with cyclic Galois group of order $n$ of a field $K$ with a primitive $n$th root of 1 and characteristic not dividing $n$ is necessarily an extension by an $n$th root}, so that there is $x\in L$ with $x^n\in K, K[x] = L$.  (We proved this last quarter in two ways, one using canonical forms of matrices, the other using Galois cohomology.)  

Along the way we learned the important Primitive Element Theorem that every finite separable extension of a field $K$ is generated by a single element over $K$.  You also derived the main facts about finite fields:  {\sl given any prime number $p$ and a power $p^n$ of it, there is a unique finite field $F_{p^n}$ of order $p^n$ up to isomorphism, which is the splitting field of $x^{p^n} - x$ over
$\Bbb Z_p$}.  It is always Galois over $\Bbb Z_p$ (and indeed over any of its subfields), with cyclic Galois group of order $n$ over $\Bbb Z_p$; this group is generated by the {\sl Frobenius automorphism} $F$ sending $x$ to $x^p$ for all $x$.  (Over a subfield of $F_{p^n}$ other than
$F_p = \Bbb Z_p$, the Galois group is generated by a suitable power of $F$, as $F$ itself will not fix all elements of the subfield.)  Every finite field takes the form $F_{p^n}$ for a unique prime power $p^n$ and $F_{p^m}$ is a subfield of $F_{p^n}$ if and only if $m$ divides $n$.  As a consequence, monic irreducible polynomials of any degree exist over any finite field, and one can even count how many of them there are without too much trouble.  

As a bonus we learned the Normal Basis Theorem (often omitted from courses covering Galois theory):  {\sl given any finite Galois extension $L$ of a field $K$ there is a basis of $L$ over $K$ consisting exactly of the conjugates of some $y\in L$ under the Galois group $G$}.  