20140205

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Today

Start by completing the division algorithm for $\mathbb{Q}[x]$. Then work on the following.

Problem (When does $(x+y)^n=x^n+y^n$)

Let $R$ be an integral domain with characteristic $n$. Compute $(x+y)^n$ for $n=1,2,3,4,5,6,7$. For which $n$ does $(x+y)^n=x^n+y^n$? Make a conjecture.

Problem 44 (Not Every Subring Is An Ideal)

Give an example of a ring $R$ and a subring $S$ so that $S$ is not an ideal of $R$. Make sure you prove that $S$ is a subring, but not an ideal.

Problem 45 (A Homomorphism From Z To A Ring With Unity)

Let $R$ be a ring with unity 1. After you have finished the first step below, you should be able to give answers to the remaining parts by referring to the first part and using the properties of ring homomorphisms.

1. Show that the mapping $\phi:\mathbb{Z}\to R$ given by $\phi(n)=n\cdot 1$ is a ring homomorphism.
2. Show that if $R$ has characteristic $n>0$, then $R$ contains a subring isomorphic to $Z_n$.
3. Show that if $R$ has characteristic $n=0$, then $R$ contains a subring that is isomorphic to $Z$.
4. Show that for any positive integer $m$, the mapping of $\phi:\mathbb{Z}\to \mathbb{Z}_m$ given by $x\to x$ mod $m$ is a ring homomorphism.

Problem 46 (Every Field Contains A Subfield Isomorphic To $\mathbb{Z}_p$ or $\mathbb{Q}$)

Suppose that $F$ is a field.

1. If $F$ has prime characteristic $p$, show that $F$ contains a subfield isomorphic to $Z_p$.
2. If $F$ has characteristic 0, show that $F$ contains a subfield isomorphic to $\mathbb{Q}$.

Problem 47 (The Remainder Theorem)

Let $F$ be a field, $a\in F$, and $f(x)\in F[x]$. Prove that $f(a)$ is the remainder in division of $f(x)$ by $x-a$.

Problem 48 (The Factor Theorem)

Let $F$ be a field, $a\in F$, and $f(x)\in F[x]$. Prove that $a$ is a zero of $f(x)$ if and only if $x-a$ is a factor of $f(x)$.

For more problems, see AllProblems