Question

Prove first statement of theorem 5.7.

Short answer

F [x] is a commutative ring with identity.

Step-by-step solution

Define addition

The addition is defined as $\left[f\left(x\right)\right]+\left[g\left(x\right)\right]:=\left[f\left(x\right)+g\left(x\right)\right]$. This gives closure to the addition.

The additive associative property directly follows from the associativity of polynomial addition:

$$\begin{gathered} \left(\left[f\left(x\right)\right]+\left[g\left(x\right)\right]\right)+\left[h\left(x\right)\right]=\left[\left(f\left(x\right)+g\left(x\right)\right)+h\left(x\right)\right] \\ =\left[f\left(x\right)+\left(g\left(x\right)+h\left(x\right)\right)\right] \\ =\left[f\left(x\right)\right]+\left(\left[g\left(x\right)\right]+\left[h\left(x\right)\right]\right) \end{gathered}$$

Similarly, the additive commutativity follows from the commutativity of polynomial addition:

$$\begin{gathered} \left[f\left(x\right)\right]+\left[g\left(x\right)\right]=\left[f\left(x\right)+g\left(x\right)\right] \\ =\left[g\left(x\right)+f\left(x\right)\right] \\ =\left[g\left(x\right)\right]+\left[f\left(x\right)\right] \end{gathered}$$

Here, the additive identity is the class of constant zero polynomial:

$$\begin{gathered} \left[f\left(x\right)\right]+\left[0\right]=\left[f\left(x\right)+0\right] \\ =\left[f\left(x\right)\right] \\ =\left[0+f\left(x\right)\right] \\ =\left[0\right]+\left[f\left(x\right)\right] \end{gathered}$$

The additive inverse is the classes of the additive inverse polynomial:

$$\begin{gathered} \left[f\left(x\right)\right]+\left[-f\left(x\right)\right]=\left[f\left(x\right)-f\left(x\right)\right] \\ =\left[0\right] \end{gathered}$$

Define multiplication

The multiplication is defined as $\left[f\left(x\right)\right]\left[g\left(x\right)\right]:=\left[f\left(x\right)g\left(x\right)\right]$. This gives closure to the multiplication.

The additive associative property directly follows from the associativity of polynomial multiplication:

$$\begin{gathered} \left(\left[f\left(x\right)\right]\left[g\left(x\right)\right]\right)\left[h\left(x\right)\right]=\left[\left(f\left(x\right)g\left(x\right)\right)h\left(x\right)\right] \\ =\left[f\left(x\right)\left(g\left(x\right)h\left(x\right)\right)\right] \\ =\left[f\left(x\right)\right]\left(\left[g\left(x\right)\right]\left[h\left(x\right)\right]\right) \end{gathered}$$

The property of distributive follows the polynomial distributivity:

$$\begin{gathered} \left[f\left(x\right)\right]\left(\left[g\left(x\right)\right]+\left[h\left(x\right)\right]\right)=\left[f\left(x\right)\left(g\left(x\right)+h\left(x\right)\right)\right] \\ =\left[f\left(x\right)g\left(x\right)+f\left(x\right)h\left(x\right)\right] \\ =\left[f\left(x\right)\right]\left[g\left(x\right)\right]+\left[f\left(x\right)\right]\left[h\left(x\right)\right] \end{gathered}$$

And

$$\begin{gathered} \left(\left[f\left(x\right)\right]+\left[g\left(x\right)\right]\right)\left[h\left(x\right)\right]=\left[\left(f\left(x\right)+g\left(x\right)\right)+h\left(x\right)\right] \\ =\left[f\left(x\right)h\left(x\right)+g\left(x\right)h\left(x\right)\right] \\ =\left[f\left(x\right)\right]\left[h\left(x\right)\right]+\left[g\left(x\right)\right]\left[h\left(x\right)\right] \end{gathered}$$

The multiplication commutativity follows from the commutativity of polynomial multiplication:

$$\begin{gathered} \left[f\left(x\right)\right]\left[g\left(x\right)\right]=\left[f\left(x\right)g\left(x\right)\right] \\ =\left[g\left(x\right)f\left(x\right)\right] \\ =\left[g\left(x\right)\right]\left[f\left(x\right)\right] \end{gathered}$$

Here, the additive identity is the class of constant zero polynomial:

$$\begin{gathered} \left[f\left(x\right)\right]\left[1\right]=\left[f\left(x\right)1\right] \\ =\left[f\left(x\right)\right] \\ =\left[1f\left(x\right)\right] \\ =\left[1\right]+\left[f\left(x\right)\right] \end{gathered}$$