Homework Assignment 6

Problems to try

Please try to solve these problems, but you do not turn them in. You can also use these results in future problems. I’m happy to talk about these problems (or any others) if you have questions.

1. Additional Problem. Let $a\in \mathbb{C}$ be nonzero, and let $b$ be some $n^\text{th}$ root of $a$ in $\mathbb{C}$ (i.e. $b$ is some solution to $x^n-a = 0$).
   1. Prove that the set of all roots in $\mathbb{C}$ of $x^n-a$ is $\{b,b\zeta_n,b(\zeta_n)^2,\ldots,b(\zeta_n)^{n-1}\}$.
   2. Conclude that $\mathbb{Q}(b,\zeta_n)$ is the splitting field for $x^n-a$ over $\mathbb{Q}$.
      
      

2. Additional Problem. Let $K$ be an algebraic extension of $F$. Prove that every subring of $K$ that contains $F$ is actually a subfield of $K$.

3. Additional Problem. Prove that $\mathbb{Q}(\zeta_3)$ is the splitting field for $x^4+x^2+1$ over $\mathbb{Q}$.

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Problems to submit

These are the problems to turn in for a grade. Please write in complete sentences, use correct punctuation, and organize your work. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Additional Problem. For each polynomial, find the splitting field over $\mathbb{Q}$ and determine the degree of the splitting field over $\mathbb{Q}$.
   1. $x^4-3$
   2. $x^4-9$
   3. $x^4-81$

   You are encouraged to use the results of the first additional problem to try. Remember that $\zeta_4 = i$.

2. Additional Problem. Let $L$ be a finite extension of $F$. Prove that if $K \subseteq L$ is a splitting field over $F$ of some (possibly infinite) set of polynomials from $F[x]$ (see the footnote¹), then $K$ is actually the splitting field over $F$ of a single polynomial.

   Here is a possible approach. First, using that $L$ is a finite extension of $F$, show that if $K$ is a splitting field for an infinite set of polynomials, then it is in fact a splitting field for a finite set of polynomials. Then take the finite set and make one big polynomial.

3. Additional Problem. Let $L = \mathbb{Q}(\sqrt[2]{2},\sqrt[3]{2},\sqrt[4]{2},\ldots)$.
   1. For $m\ge 2$, let $L_m = \mathbb{Q}(\sqrt[2]{2},\sqrt[3]{2},\sqrt[4]{2},\ldots,\sqrt[m]{2})$. Prove that $L = \bigcup_{m\ge 2}L_m$.
   2. Prove that $L$ is an algebraic extension of $\mathbb{Q}$.
   3. Prove that $[L:\mathbb{Q}]$ is infinite.

   For part c, consider explicity showing $L$ contains a subfield of degree $k$ over $\mathbb{Q}$ for each number $k$.
   

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Footnotes