~~META:title=Earlier Homework~~

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===== Math 504 - Earlier Homework =====
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**Problem Set 6** Due 04/14/2020 (complete)
  - Show that every $\varphi\in\aut_K\left(\ol{K}\right)$ induces a complete lattice automorphism on $\sub_K\left(\ol{K}\right)$. (This is part of Prop. 4.6.3 in the posted class notes)
  - Let $E/K$ be an algebraic extension.  Prove that the normal closure of $E/K$ is the spliting field of the set of polynomials  \[ A = \left\{{\ds\min_K(\alpha)}\mid\alpha\in E^\times\right\}. \] (This is Prop. 4.7.3 in the posted class notes)
  - Let $E/K$ be an infinite separable extension. Prove that $[E:K]_s =_f [E:K]$, meaning both are finite and equal, or both are infinite.  (Note that this and its converse were already proved for finite extensions as Prop. 3.71.3 in the posted class notes) 
  - Find an example of an algebraic extension for which $[E:K]_s =_f [E:K]$, but $E/K$ is NOT separable.

**Problem Set 5** Due 03/24/2020 (complete)
  - Let $K\leq E\leq F$, and $\alpha\in F$, algebraic over $K$. Prove: 
    - If $\alpha$ is separable over $K$, then it is separable over $E$.
    - If $\alpha$ is separable over $E$, and $E/K$ is separable, then $\alpha$ is separable over $K$.
  - Let $K=\F_2(s,t)$ be the field of rational functions in two variables $s$ and $t$, over the two element field, $\F_2$.  Let $\alpha=\sqrt{s}$ and $\beta=\sqrt{t}$, i.e. $\alpha$ is a root of $x^2-s\in K[x]$, and similarly for $\beta$.  Prove or disprove that $K(\alpha,\beta)$ is a simple extension of $K$. 
  - Let $K$ be a field of characteristic $p$. 
    - Show that $K=K^{1/p}$ iff $K$ is perfect.
    - Show that the field $K^{1/p^\infty}$ is a perfect field, and the smallest perfect field that contains $K$. 
  - Let $K$ be a field of characteristic $p$. Is $\ol{K}$ separable over $K^{1/p^\infty}$? Prove or disprove.


**Problem Set 4** Due 03/10/2020(complete)
  - Show that the algebraic closure is a //closure operator//, i.e.
    - $K\leq\ol{K}$, \\ \\  
    - $\ol{\ol{K}} =\ol{K}$, \\
    - $K\leq E \imp \ol{K}\leq\ol{E}$. \\
  - Let $\ol{K}$ be an algebraic closure of $K$.  Show:
    - $\ol{K}$ is minimal with the property of being an extension of $K$ which is algebraically closed.
    - $\ol{K}$ is maximal with the property of being an algebraic extension of $K$.
  - Let $\ f(x)\in K[x]$. Prove that if $\alpha$ is a root of $\ f(x)\ $ with multiplicity $m$, then $\alpha$ is a root of $\ f^{(i)}(x)\ $ for all $\ 0\leq i < m$.
  - Prove that if $K$ is a perfect field, and $F/K$ is an algebraic extension, then $F$ is a perfect field.


**Problem Set 3** Due 02/25/2020 (complete)
  - Prove the corollary stated in class: If $K\leq E_i\leq F$ and each $E_i/K$ is algebraic, then the join $\displaystyle\bigjoin_{i\in I}E_i$ is algebraic over $K$.
  - Let $F/K$ be a finite extension.  Prove that $\end_K(F)=\aut_K(F)$, i.e. every endomorphism of $F$ that fixes $K$ is an automorphism of $F$.
  - Consider the extension $F=\Q(\alpha,\omega)$ of $\Q$ discussed in class, where $\alpha$ is a root of $x^3-2$ and $\omega$ is a root of $x^2+x+1$. Construct several automorphisms of $F$.  Is there a bound for the number of automorphisms of $F$?
  - Let $F/K$ be a field extension, and $\varphi:F\to L$ a field homomorphism. Let $\widehat{F}=\varphi(F)$ and $\widehat{K}=\varphi(K)$. Prove:
    - $[\widehat{F}:\widehat{K}]=[F:K]$.
    - If $F/K$ is algebraic, then so is $\widehat{F}/\widehat{K}$.
    - If $F/K$ is transcendental, then so is $\widehat{F}/\widehat{K}$.
    - If $F$ is an algebraic closure of $K$, then $\widehat{F}$ is an algebraic closure of $\widehat{K}$.

**Problem Set 02** Due 02/13/2020 (complete)
  -  Let $F$ be a field extension of $K$, and $a_1,a_2,\dots,a_n\in F$.  Prove:
    - $ K[a_1,a_2,\dots,a_n] = K[a_1][a_2]\cdots[a_n]$, and
    - $ K(a_1,a_2,\dots,a_n) = K(a_1)(a_2)\cdots(a_n)$.
  - Show that $\Q(\sq{2})\neq\Q(\sq{3})$. Generalize.
  - Grillet, Page 163, IV.2.1 
  - Grillet, Page 163, IV.2.2, IV.2.4


**Problem Set 01** Due 02/04/2020 (complete)
  - Let $G$ be a group and $N\normaleq G$. $G$ is solvable iff $N$ and $G/N$ are solvable. In this case, $$l(G) \leq l(N) + l(G/N).$$
  - If $L$ is a poset in which every subset has a l.u.b., then every subset of  $L$ also has a g.l.b.
  - Given a lattice $(L,\join,\meet)$ in the algebraic sense, show that the binary relation $\leq$, defined by $$ x \leq y \quad iff \quad x \meet y = x, $$ is a partial order on $L$. Moreover, for any $x,y \in L$, $x \meet y$ is the $\glb\{x,y\}$, and $x \join y$ is the $\lub\{x,y\}$. 
  - Let $A$ be a universal algebra, and $\sub(A)$ the complete lattice of subuniverses of $A$. If $D\subseteq\sub(A)$ is directed, then $\ds\left(\bigunion_{X\in D}X\right)\in\sub(A)$.



[[people:fer:504ws:Spring2020:homework]]