Finitely generated field extension
Imagine that numbers are like toys, and you want to play with them. But sometimes, you don't have enough toys to play with, so you need to borrow some from your friend.
In math, we have something called a "field," which is like a collection of numbers that you can add, subtract, multiply, and divide. It's a special kind of place where you can do all these fun operations.
Sometimes, we want to make our field even bigger by adding more numbers to it. This is called a "field extension." It's like your toy collection getting bigger because your friend gave you some new toys to play with.
Now, let's say that your friend also has a limited number of toys. In other words, they have a finite toy collection. When they give you some toys, you can use them to make your collection bigger, but you still have a limited number of toys overall.
In math, we say that a "finitely generated field extension" is when we add some new numbers to our field, but these new numbers can be created by using a limited number of specific numbers. It's like your friend gave you some special toys, and with those special toys, you can create all the other toys you want.
For example, let's say your toy collection is like a field, and you have numbers 1, 2, and 3 in it. But now your friend gives you one more special toy, like number 5. And guess what? You can create every toy you want just using these numbers: 1, 2, 3, and 5. So your toy collection just became a finitely generated field extension!
It's important to note that not all field extensions are finitely generated. Sometimes, your friend might give you an infinite number of new toys, and you can never create all those toys using just a limited set of numbers. That would be an example of a field extension that is not finitely generated.
So, a finitely generated field extension is when we can add some new numbers to our field, but those numbers can be created using a limited number of specific numbers. It's like making our toy collection bigger, but still having a finite number of toys overall.
In math, we have something called a "field," which is like a collection of numbers that you can add, subtract, multiply, and divide. It's a special kind of place where you can do all these fun operations.
Sometimes, we want to make our field even bigger by adding more numbers to it. This is called a "field extension." It's like your toy collection getting bigger because your friend gave you some new toys to play with.
Now, let's say that your friend also has a limited number of toys. In other words, they have a finite toy collection. When they give you some toys, you can use them to make your collection bigger, but you still have a limited number of toys overall.
In math, we say that a "finitely generated field extension" is when we add some new numbers to our field, but these new numbers can be created by using a limited number of specific numbers. It's like your friend gave you some special toys, and with those special toys, you can create all the other toys you want.
For example, let's say your toy collection is like a field, and you have numbers 1, 2, and 3 in it. But now your friend gives you one more special toy, like number 5. And guess what? You can create every toy you want just using these numbers: 1, 2, 3, and 5. So your toy collection just became a finitely generated field extension!
It's important to note that not all field extensions are finitely generated. Sometimes, your friend might give you an infinite number of new toys, and you can never create all those toys using just a limited set of numbers. That would be an example of a field extension that is not finitely generated.
So, a finitely generated field extension is when we can add some new numbers to our field, but those numbers can be created using a limited number of specific numbers. It's like making our toy collection bigger, but still having a finite number of toys overall.