189-457B: Algebra 4

Assignment 4

Due: Wednesday, March 26.

1. Show that if $E_1$ and $E_2$ are two finite extensions of a field $F$ whose degrees $[E_1:F]$ and $[E_2:F]$ are relatively prime to each other, then $E_1\cap E_2=F$.

2.

(a) Show that there are $2$ (respectively, $1$, $2$, $3$, and $6$) distinct irreducible polynomials of degree $1$ (respectively, $2$, $3$, $4$, and $5$) over the field $\mathbb{F}_2=\{0,1\}$ with two elements, and write these polynomials down. (Hint: the only case that could be a bit tedious is for degree $5$, but it is not so bad if you start off on the right foot. Excluding polynomials with a root in $\mathbb{F}_2$, the suspects are all of the form $x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 1$ with an odd number of $a_i's$ equal to $1$. Now eliminate the polynomials that are the product of an irreducible cubic and quadratic polynomial and see what you are left with.)

(b) Construct a field ${\mathbb F}_{32}$ with $32$ elements and show that it contains no proper subfields other than the field with two elements. Conclude that every element of ${\mathbb F}_{32}-{\mathbb F}_2$ is the root of an irreducible polynomial of degree $5$ over ${\mathbb F}_2$.

(c) Conclude from your work in (a) and (b) that any two fields of cardinality $32$ are necessarily isomorphic.

3. Let $\mathbb F_p$ denote the finite field with $p$ elements. Show that the difference of any two roots of the polynomial $f(x) = x^p-x-1$ (in a splitting field) belongs to $\mathbb F_p$. Conclude that $f(x)$ is irreducible over $\mathbb F_p$. (Caveat: This is stronger than merely showing that $f(x)$ has no roots over $\mathbb F_p$...). Describe the splitting field of $f(x)$, and write down its degree over $\mathbb F_p$. What is the Galois group of $f(x)$ over $\mathbb F_p$?

4. Describe the splitting field and the Galois group $G$ of the quintic polynomial $x^5-3$ over $\mathbb Q$. What is the cardinality of $G$? Does it correspond to a familiar subgroup of $S_5$ that you have already encountered?

5. Let $F$ be an infinite field of characteristic $p$. (For instance, the field $F= \mathbb F_p(t)$ of rational functions in an indeterminate $t$, with coefficients in the finite field with $p$ elements, is a good example to have in mind.) A polynomial $f(x) \in F[x]$ is said to be linear if it is of the form $$f(x) = a_n x^{p^n} + a_{n-1} x^{p^{n-1}} + \cdots + a_1 x^p + a_0 x, \qquad a_0, \ldots, a_n \in F,$$ i.e., if the degrees of all the non-zero monomials that appear in $f(x)$ are powers of $p$. Show that a linear polynomial satisfies $$f(a+b) = f(a) + f(b), \quad f(\lambda a) = \lambda f(a), \qquad \mbox{ for all } \ a,b\in F, \ \ \lambda\in \mathbb F_p,$$ (justifying the terminology). Show that the Galois group of $f(x)$ is isomorphic to a subgroup of ${\rm GL}_n(\mathbb F_p)$.

Cultural remark. It follows in particular that the Galois group of the polynomial $x^8 + x^2 + t x$ over the field $F=\mathbb F_2(t)$ is contained in one of our objects of predilection, the group $\mathbf{GL}_3(\mathbb F_2)$ of order $168$. A theorem of Abhyankar asserts that the Galois group of this polynomial is in fact equal to $\mathbf{GL}_3({\mathbb F}_2)$. More generally, the group $G=\mathbf{GL}_n(\mathbb F_2)$ is the Galois group of the polynomial $x^{2^n} + x^2 + tx$ over $\mathbb F_2(t)$.

6. Let $E$ be the splitting field of the polynomial $x^p-t$ over the field $F = \mathbb F_p(t)$. Describe its degree, and show that ${\rm Aut}(E/F)$ is the trivial group (consisting only of the identity.)

Cultural remark. The field $E$ is an example (and in some sense, a very prototypical one) of a splitting field which fails to be Galois, because it is inseperable. Such examples never arise over fields of characteristic zero, where every splitting field (over a field $F$ of characteristic zero) has as many automorphisms as its degree over $F$.

7. Let $F$ be a field of characteristic zero, and let $F(t)$ be the field (of infinite degree over $F$) of rational functions over $F$ in the indeterminate $t$.

(a) Show that $F(t)$ has finite degree over any subfield which properly contains $F$.

(b) Let $G:= {\rm Aut}(F(t)/F)$. Show that $G$ contains involutions (elements of order $2$) $\sigma$ and $\tau$ which fix $t^2$ and $t^2-t$ respectively, and for which $\sigma\tau$ is of infinite order.

(c) Use the results of (a) and (b) combined with the Galois correspondence to show that any rational function in $t$ that is expressible both as a rational function in $t^2$ and in $t^2-t$ is a constant; i.e., that $F(t^2) \cap F(t^2-t) = F$.

8. Let $p$ be a prime number and let $F/\mathbb Q$ be the field generated by a primitive $p$-th roots of unity $\zeta= e^{2\pi i /p}$. Recall that we have shown that the Galois group of $F/\mathbb Q$ is (canonically) isomorphic to the multiplicative group $G = \mathbb F_p^\times$. Let $H \subset G$ be the subgroup of non-zero quadratic residues in $G$. Show that the elements $$ \alpha = \sum_{j\in H} \zeta^j = \sum_{j=1}^{(p-1)/2} \zeta^{j^2}, \qquad \beta = \sum_{j\in G-H} \zeta^j$$ generate a quadratic extension of $\mathbb Q$ and that they are Galois conjugates of each other. Conclude from this that $\alpha+\beta$ and $\alpha\beta$ are rational numbers, and compute their values. Use this to express $\alpha$ and $\beta$ in the form $a\pm b\sqrt{d}$ with $a,b,d\in \mathbb Q$.

9. Let $f(x)$ be a degree $n\ge 5$ polynomial over a field $F$ whose Galois group is isomorphic to the full permutation group $S_n$. If $n_1+\cdots + n_k=n$ is any partition of $n$, show that there is an extension $E$ of $F$ for which the polynomial $f(x)$ factors into irreducible polynomials of degrees $n_1,\ldots, n_k$ in the larger polynomial ring $E[x]$.

10. Let $f(x)$ be an irreducible septic polynomial with coefficients in a field $F$, and assume that its Galois group is $GL_3(\mathbb F_2)$ (like the example obtained in problem 5.) Show that the splitting field $E$ of $f(x)$ contains two subfields that are of degree $7$ over $F$ and are not isomorphic as fields.