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189-570A: Higher Algebra I

Assignment 5

Due: Monday, November 13.

1. Let $K/F$ be a finite Galois extension and let $L/F$ be an arbitrary finite extension. Assume that $K$ and $L$ are contained in a common extension ${\bar F}$ of $F$, and recall that the compositum $KL$ is the smallest subfield of ${\bar F}$ that contains both $K$ and $L$. Show that $KL$ is a Galois extension of $L$, that $K$ is a Galois extension of $K\cap L$, and that there is a natural isomorphism $$ {\rm Gal}(KL/L) \rightarrow {\rm Gal}(K/K\cap L).$$

2. Let $K_1/F$ and $K_2/F$ be two Galois extension with Galois groups $G_1$ and $G_2$, which are contained in a common field ${\bar F}$. Show that the compositum $K_1K_2/F$ is also a Galois extension, and that its Galois group is naturally identified with a subgroup of $G_1\times G_2$. Describe this subgroup precisely.

3. Let $F$ be a field of characteristic $\ne 2$ and let $K = F(\sqrt{a})$ be a quadratic extension of $F$. Show that there is a cyclic extension $L/F$ of degree $4$ containing $K$ if and only if $a$ can be written as a sum of two squares in $F$. Conclude that the quadratic field ${\mathbb Q}(\sqrt{5})$ can be embedded in a cyclic quartic extension of $\mathbb Q$, while $\mathbb Q(\sqrt{-5})$ can not. Use the theory of cyclotomic fields to give a simple example of a cyclic quartic field containing $\mathbb Q(\sqrt{5})$.

4. Let $p_1$, $p_2$, $p_3$, $\ldots$ $p_n$ be distinct prime numbers. Show that $K_n=\mathbb Q(\sqrt{p_1}, \ldots, \sqrt{p_n})$ is a Galois extension of $\mathbb Q$, and that its Galois group is isomorphic to an $n$-dimensional vector space over the field ${\bf F}_2$ with two elements.

5. Let $p_1$,$p_2$, $\ldots$, be an infinite sequence of prime numbers, and (with notations as in the previous problem) let $K_\infty:= \cup_n K_n$ (where the union is taken in a fixed algebraic closure of $\mathbb Q$, say).

(a) Show that $K_\infty$ is countable, and that the set of finite extensions of $\mathbb Q$ in $K_\infty$ is countable as well.

(b) Show that the group $G={\rm Gal}(K_\infty/K)$ is uncountable and has uncountably many distinct subgroups of finite index (in fact, of index two). (Hint: you have to use the axiom of choice! One common form of the axiom of choice is the assertion that every vector space has a basis.)

Remark: This shows that a naive Galois correspondence could not possibly work for $K_\infty/\mathbb Q$, since there are a lot more subgroups of $G$ of index two than there are quadratic fields contained in $K_\infty$.

6. What are the Galois groups of the following polynomials?

(a) $x^3-x-1$ over $\mathbb Q$.

(b) $x^3-x-1$ over $\mathbb Q(s)$, where $s^2=-23$.

(c) $x^4 -5$ over $\mathbb Q$. Over $\mathbb Q[t]/(t^2-5)$? Over $\mathbb Q[t]/(t^2+5)$? Over $\mathbb Q[t]/(t^2+1)$?

(d) $x^n-t$, over $\mathbb C(t)$.

(e) $x^4-t$, over $\mathbb R(t)$.

7. Show that the fractional linear transformation $\sigma$ defined by $\sigma(x) = \frac{1}{1-x}$ is of order $3$, and construct an element $t$ which generates the subfield $\mathbb Q(x)^{\sigma=1}\subset \mathbb Q(x)$ of rational functions which are invariant under $\sigma$. Use this to write a polynomial $P(x,t)\in \mathbb Q(t)[x]$ of degree three in $x$ whose Galois group over $\mathbb Q(t)$ is $\mathbb Z/3\mathbb Z$.

8. Letting $P(x,t)$ be the polynomial obtained in question 7, show that the Galois group of $P(x,a)$ over $\mathbb Q$ is a subgroup of $\mathbb Z/3\mathbb Z$ for all $a\in\mathbb Q$.

Optional supplement. If you have access to a computer algebra package like PARI, compute the Galois group of $P(x,a)$ for various values of $a$ chosen at random (say, if $a$ an integer between $1$ and $10$, for example) using the polgalois command. What do you observe?

9. Let $f$ be a polynomial over $\mathbb Q$ of degree $n$, and let $K/\mathbb Q$ be its splitting field. Suppose that ${\rm Gal}(K/\mathbb Q)$ is isomorphic to the symmetric group $S_n$ with $n > 2$.

(a) Show that $f$ is irreducible over $\mathbb Q$.

(b) If $a$ is a root of $f$, show that the only automorphism of $\mathbb Q(a)$ is the identity.

(c) If $n > 3$, show that $a^n$ does not belong to $\mathbb Q$.

10. Let $k$ be the finite field with $p$ elements ($p$ a prime). Let $K=k(t)$ be the field of rational functions in the variable $t$. Let $G$ be the group of automorphisms of $K$ obtained by the mappings $$t \mapsto \frac{at + b}{ct+d}, \quad \mbox{with } a,b,c,d \in k, \ \ ad-bc\ne 0$$ that you already encountered in your previous problem set. Recall that in that problem set, you showed that the field $K^B$ fixed by the Borel subgroup of $G$ (consisting of upper triangular matrices) is generated by the element $y = (x^p-x)^{p-1}$. Show that $$ w:= \gamma_1 y + \gamma_2 y + \cdots \gamma_{p+1} y$$ belongs to $K^G$, where the elements $\gamma_j$ are representatives for the coset space $B\backslash G$: $$ G = B\gamma_1 \cup \cdots \cup B\gamma_{p+1}.$$ Use this to calculate $y$ explicitly, and show that it generates $K^G$.