Ruby on Rails: Categories not working or sorting

The posts and categories are not agreeing on something and I'm not sure what it is. What am I doing wrong?

As soon as I click a category nothing changes but the url to http://localhost:3000/posts?category=Physics

class Post < ActiveRecord::Base
  belongs_to :category
  # ...
end

class Category < ActiveRecord::Base
  belongs_to :posts
  # ...
end

Here is the Posts Controller:

class PostsController < ApplicationController     

    def index
        if 
            params[:category_id].blank?
            @posts = Post.all.order("created_at DESC")
        else
            @category_id = Category.find_by(name: params[:category]).id
            @posts = Post.where(category_id: @category_id).order("created_at DESC")
        end
    end

# ...
    def post_params
        params.require(:post).permit(:title, :link, :description, :category_id)
    end

Here is the index file:

<% Category.all.each do |category| %>
    <%= link_to category.name, posts_path(category: category.name) %>
    <% end %>


          <% @posts.each do |post| %>
              <%= link_to post.title, post  %>
            <%= link_to time_ago_in_words(post.created_at) + " ago", post %>  
          <% end %>

This is the migration for Posts:

  class CreatePosts < ActiveRecord::Migration
  def change
    create_table :posts do |t|
      t.string :title
      t.string :link
      t.text :description

      t.timestamps null: false
    end
  end
end

This is the migration for AddIdToPosts

class AddIdToPosts < ActiveRecord::Migration
  def change
      add_column :posts, :category_id, :integer
  end
end

This is what console told me about the last post:

irb(main):008:0> Category

=> Category(id: integer, name: string, created_at: datetime, updated_at: datetime)

irb(main):009:0> @post = Post.last

  Post Load (17.0ms)  SELECT  "posts".* FROM "posts"  ORDER BY "posts"."id" DESC LIMIT 1

=> #<Post id: 3, title: "Testing", link: "", 
       description: "testing", created_at: "2015-05-06 07:08:36", 
       updated_at: "2015-05-06 07:08:36", user_id: 1, category_id: 1>