MHB Find p-adic valuation and p-norm

[evinda]

Hi! (Nerd)

I have to find the values $w_p(x)$ and $|x|_p$ for $x=\frac{3}{686}$ and $x=\frac{56}{12}$ for $p=7,5$.

That's what I have tried:

• $$x=\frac{3}{686}=\frac{3}{2 \cdot 7^3}=3 \cdot 2^{-1} \cdot 7^{-3} $$
  
  $$p=7: x=7^{-3} \cdot 3 \cdot 2^{-1} \mapsto 7^3=343= \left |\frac{3}{686} \right|_7$$
  
  $$w_7 \left ( \frac{3}{686}\right)=-3$$
  
  $$p=5: x=5^0 \cdot 7^{-3} \cdot 3 \cdot 2^{-1} \mapsto 5^{-0}=1=\left |\frac{3}{686} \right |_5$$
  
  $$w_5 \left ( \frac{3}{686}\right)=0$$
  $$$$
• $$x=\frac{56}{12}=2 \cdot 7 \cdot 3^{-1} $$
  
  $$p=7: x=7^1 \cdot 2 \cdot 3^{-1} \mapsto 7^{-1}=\frac{1}{7}=\left |\frac{56}{12} \right |_7$$
  
  $$w_7 \left ( \frac{56}{12}\right)=1$$
  
  $$p=5: x=5^0 \cdot 2 \cdot 7 \cdot 3^{-1} \mapsto 5^{-0}=1=\left |\frac{56}{12} \right |_5$$
  
  $$w_5 \left ( \frac{56}{12}\right)=0$$
  

Could you tell me if it is right or if I have done something wrong? (Thinking)

 

[mathbalarka]

Looks good to me. (Yes)

 

[evinda]

Looks good to me. (Yes)

Nice! Thanks a lot! (Smile)