# Field Theory Qual Review

# Robert Won Prof. Rogalski

## 1 (Some) qual problems

• (Fall 2007, 5) Let F be a field of characteristic p and $f \in F [ x ]$ a polynomial $\textstyle f ( x ) = \sum _ { i } f _ { i } x ^ { i }$ Give necessary and sufficient conditions on the $\{ f _ { i } \}$ for $f ( x ^ { p } )$ to itself be a $p ^ { \mathrm { t h } }$ power, i.e. $\exists g ( x )$ such that $f ( x ^ { p } ) = g ( x ) ^ { p }$ . In particular, prove that your condition is necessary.

• (Fall 2007, 6) Let $F / K$ be a field extension of degree 2

a. If K is characteristic not 2, show that $F / K$ is Galois.

b. Give an example where $F / K$ is Galois even though char $K = 2$

c. Give an example where $F / K$ is not Galois.

• (Fall 2009, 3) Let F be a finite field of order q and $E / F { \mathrm { ~ a ~ } }$ field extension. Suppose that an element $a \in E$ is algebraic over F . Prove that $[ F ( a ) : F ]$ is the smallest positive integer n such that $a ^ { q ^ { n } } = a$ and that it divides every other such positive integer.

• (Fall 2009, 4) Let G be any finite group and F any field. Show that there exist fields L and E with $F \subseteq L \subseteq E$ , such that E is Galois over L with the Galois group of $E / L$ being isomorphic to G.

• (Fall 2009, 5) Consider the splitting field of E of the polynomial $f ( x ) = x ^ { 4 } - 5$ over $\mathbb { Q }$

a. Find the degree $\left[ E : \mathbb { Q } \right]$

b. Determine the Galois group of E over $\mathbb { Q }$ as a subgroup of $S _ { 4 }$

• (Spring 2008, 4) Suppose that there exists an intermediate field L of the Galois extension $F / E$ of degree 2 over E. What can we say about $\mathrm { G a l } ( F / E ) ?$

• (Spring 2009, 4) Let $a = { \sqrt { 2 + { \sqrt { 2 } } } }$ in C and let f be the minimal polynomial of a over $\mathbb { Q } .$ Let E be the splitting field for f over Q. Determine the Galois group $\operatorname { G a l } ( E / \mathbb { Q } )$ .

• (Spring 2009, 5) Let $E / F$ be a Galois extension and let K, L be intermediate fields. Show that K and L are F -isomorphic (i.e. there exists an isomorphism from K to L which is the identity on F ) if and only if hte subgroups of $G = \operatorname { G a l } ( E / F )$ corresponding to K and L are conjugate in G.

## 2 (Some) field things to know

Throughout, F and K are fields.

• Basic facts and definitions. (characteristic, prime subfield, field extension, degree of a field extension, field extensions generated by elements, primitive elements, algebraic extensions)

• The characteristic of F is either 0 or prime.

• Any homomorphism of fields is 0 or injective.

• Let $p ( x ) \ \in \ F [ x ]$ be irreducible. Then there exists a field extension $K / F$ in which $p ( x )$ has a root. In particular, $K = F [ x ] / p ( x )$ and $[ K : F ] = n$ . If $\deg p ( x ) = n$ and $\theta = x$ mod $( p ( x ) ) \in K$ then $1 , \theta , \ldots , \theta ^ { n - 1 }$ are an F -basis for $K$

• Let $p ( x ) \in F [ x ]$ be irreducible. If K is an extension of F containing α a root of $p ( x )$ then $F ( \alpha ) \cong F [ x ] / p ( x )$

• Let $\varphi : F  F ^ { \prime }$ be an isomorphism of fields and $p ( x ) \in F [ x ]$ be irreducible. Let $p ^ { \prime } ( x ) \in F ^ { \prime } [ x ]$ be the irreducible polynomial obtained by applying $\varphi$ to the coefficients. Let α be a root of $p ( x )$ and $\beta$ be a root of $p ^ { \prime } ( x )$ . Then there is an isomorphism

$$
\sigma : F ( \alpha )  F ^ { \prime } ( \beta )
$$

such that $\sigma ( \alpha ) = \beta$ and $\sigma | _ { F } = \varphi$

• Let α be algebraic over F . Then there is a unique monic irreducible polynomial $m _ { \alpha , F } ( x ) \in$ $F [ x ]$ which has α as a root. The polynomial $m _ { \alpha , F } ( x )$ is called the minimal polynomial and its degree is called the degree of α.

• If $L / F$ is an extension of fields and α is algebraic over F and L then $m _ { \alpha , L } ( x )$ divides $m _ { \alpha , F } ( x )$ in L.

• Let α be algebraic over F , then $F ( \alpha ) \cong F [ x ] / ( m _ { \alpha } ( x ) )$ and $\left[ F ( \alpha ) : F \right] = \deg m _ { \alpha } ( x ) = \deg \alpha$

• The element α is algebraic over F if and only if $F ( \alpha ) / F$ is finite.

• If $K / F$ is finite, then it is algebraic.

• If $F \subseteq K \subseteq L$ are fields then $[ L : F ] = [ L : K ] [ K : F ]$

• The extension $K / F$ is finite if and only if K is generated by a finite number of algebraic elements over F .

• If α and $\beta$ are algebraic over F then $\alpha \pm \beta , \alpha \beta , \alpha / \beta$ are all algebraic.

• Let $L / F$ be an arbitrary extension. Then the collection of elements of L that are algebraic over F form a subfield K of L.

• If K is algebraic over F and L algebraic over K then L is algebraic over $F .$ .

• Let $K _ { 1 }$ and $K _ { 2 }$ be two finite extensions of a field F contained in K. Then

$$
[ K _ { 1 } K 2 : F ] \le [ K _ { 1 } : F ] [ K _ { 2 } : F ]
$$

with equality if and only if an F -basis for one of the fields remain linearly independent over the other field.

• Splitting fields exist and the splitting field of a polynomial is unique up to isomorphism.

• If K is an algebraic extension of F which is the splitting field over F for some collection of polynomials, then K is called a normal extension of F .

• A splitting field of a polynomial of degree n has degree at most n!.

• A polynomial $f ( x )$ has a multiple root α if and only if α is also a root of its derivative. In particular, $f ( x )$ is separable if and only if it is relatively prime to its derivative.

• Every irreducible polynomial over a field of characteristic 0 or a finite field is separable.

• If char $F = p$ then $( a + b ) ^ { p } = a ^ { p } + b ^ { p }$ and $( a b ) ^ { p } = a ^ { p } b ^ { p }$

• Let $p ( x )$ be an irreducible polynomial over F a field of characteristic $p .$ Then there exists a unique integer $k \geq 0$ and a unique irreducible separable polynomial $p _ { \mathrm { s e p } } ( x ) \in F [ x ]$ such that

$$
p ( x ) = p _ { \mathrm { s e p } } ( x ^ { p ^ { k } } ) .
$$

• Every finite extension of a perfect field is separable.

• Cyclotomic polynomials: Let $\zeta _ { n }$ be a primitive $n ^ { \mathrm { t h } }$ root of unity.

The $n ^ { \mathrm { t h } }$ cyclotomic polynomial $\Phi _ { n } ( x )$ is the degree $\varphi ( n )$ polynomial whose roots are the primitive $n ^ { \mathrm { t h } }$ roots of unity:

$$
\Phi _ { n } ( x ) = \prod _ { \zeta { \mathrm { p r i m i t i v e } } } ( x - \zeta ) = \prod _ { ( a , n ) = 1 } ( x - \zeta _ { n } ^ { a } ) .
$$

$\Phi _ { n } ( x )$ is a monic polynomial in $\mathbb { Z } [ x ]$ which is the unique irreducible monic polynomial of degree $\varphi ( n )$ .

$$
x ^ { n } - 1 = \prod _ { d \mid n } \Phi _ { d } ( x )
$$

$$
\Phi _ { p } ( x ) = x ^ { p - 1 } + x ^ { p - 2 } + \cdot \cdot \cdot + x + 1 .
$$

• Let $K / F$ be a field extension and $\alpha \in K$ algebraic over F . Then for any $\sigma \in \operatorname { A u t } ( K / F )$ , σ α is a root of the minimal polynomial for α over $F ;$ that is $\operatorname { A u t } ( K / F )$ permutes the roots of irreducible polynomials

$| \operatorname { A u t } ( E / F ) | \leq [ E : F ]$

• Galois extensions: $K / F$ is Galois if any of the following equivalent conditions hold

(1) $K / F$ is a splitting field of a collection of separable polynomials over F

(2) F is the precisely the set of elements fixed by $\operatorname { A u t } ( K / F )$ (in general, the fixed field may be larger than F

(3) $[ K : F ] = | \operatorname { A u t } ( K / F ) |$

(4) $K / F$ is finite, normal, and separable

• (Fundamental Theorem of Galois Theory) Let $K / F$ be a Galois extension and $G = { \mathrm { G a l } } ( K / F$ Then there is a bijection

$$
\{ { \mathrm { s u b f i e l d s ~ } } E { \mathrm { ~ o f ~ } } K { \mathrm { ~ c o n t a i n i n g ~ } } F \} \longleftrightarrow \{ { \mathrm { s u b g r o u p s ~ } } H { \mathrm { ~ o f ~ } } G \}
$$

given by the correspondence

$$
E  \{ \mathrm { t h e ~ e l e m e n t s ~ o f ~ } G \mathrm { ~ f i x i n g ~ } E \}
$$

{the fixed field of $H \}  H$

which are inverse. Under this correspondence,

(1) $E _ { 1 } \subseteq E _ { 2 }$ if and only if $H _ { 2 } \leq H _ { 1 }$

(2) $[ K : E ] = | H |$ and $[ E : F ] = | G : H |$

(3) $K / E$ is Galois with Galois group H

(4) $E / F$ is Galois if and only if H is normal. In this case, the Galois group of $E / F$ is $G / H$

(5) The intersection $E _ { 1 } \cap E _ { 2 }$ corresponds to the group $\langle H _ { 1 } , H _ { 2 } \rangle$ and the composite field $E _ { 1 } E _ { 2 }$ corresponds to $H _ { 1 } \cap H _ { 2 }$ •

• Any finite field is isomorphic to $\mathbb { F } _ { p ^ { n } }$ which is the splitting field over $\mathbb { F } _ { p }$ of the polynomial $x ^ { p ^ { n } } - x .$ , with cyclic Galois group of order n generated by the Frobenius automorphism $\sigma _ { p }$ The subfields of $\mathbb { F } _ { p ^ { n } }$ are the fields $\mathbb { F } _ { p ^ { d } }$ and are all Galois over $\mathbb { F } _ { p } ,$ they are the fixed fields of $\sigma _ { p } ^ { d }$ for $d \mid n$

• The finite field $\mathbb { F } _ { p ^ { n } }$ is simple.

• The polynomial $x ^ { p ^ { n } } - x$ is the product of all the distinct irreducible polynomials in $\mathbb { F } _ { p } [ \boldsymbol { x } ]$ of degree d where d runs across the divisors of n.

• The Galois group of the cyclotomic field $\mathbb { Q } ( \zeta _ { n } )$ of $n ^ { \mathrm { t h } }$ roots of unity is isomorphic to the multiplicative group $( \mathbb { Z } / n \mathbb { Z } ) ^ { \times }$ . The isomorphism is given by

$$
\begin{array} { r l } & { ( \mathbb { Z } / n \mathbb { Z } ) ^ { \times } \to \operatorname { G a l } ( \mathbb { Q } ( \zeta _ { n } ) / \mathbb { Q } ) } \\ { a } & { ( \mathrm { m o d ~ } n ) \mapsto \sigma _ { a } } \end{array}
$$

where $\sigma _ { a } ( \zeta _ { n } ) = \zeta _ { n } ^ { a }$

• The extension $K / F$ is called abelian if $K / F$ is Galois and $\operatorname { G a l } ( K / F )$ is abelian.

• If G is any finite abelian group, then there is a subfield K of the a cyclotomic field with ${ \mathrm { G a l } } ( K / \mathbb { Q } ) \cong G$