What does the idea of splitting mean when used with fields and polynomials?

Question

i want to understand what does field splitting represent,from my book

A Course In Galois Theory by D.J.H Garling

this term is explained by following sentences

Suppose that $K$ is field, that $f\in K[x]$ and that $L:K$ is an extension,we say that $f$ splits over $L$ if we can write

$f=\lambda(x-\alpha_1)*....*(x-\alpha_n)$

where $\alpha_1,...\alpha_n$ are in $L$ and $\lambda$ $\in$ $K$

does it means that field splitting is simple procedure of concatenation of for example real numbers and complex numbers to get some result?i meant when we are talking about filed splitting we simple take element from one field and another from some other field and concatenate using operation multiply,divide to get result what we want?it may looks as a primitive question,but i want to know general idea behind this terminology ,thanks in advance

Answer 1

The word "splitting" refers historically to the splitting of a polynomial into linear factors. If we have an irreducible polynomial with coefficients in a field $K$, it may split into linear factors when considered within some larger field $L$.

For example, the irreducible Real polynomial $x^2+1$ splits over the Complex numbers as $(x+i)(x-i)$.The Real numbers can be regarded as a subfield of the Complex numbers, or the Complex numbers as an extension of the Reals.

With a little care about conditions - but in a wide range of useful situations - it can be proved that IF $p(x)$ is an irreducible polynomial [which means it has no non-trivial factors, not just no linear factors] over some field $F$, and $p(x)$ splits into linear factors in some larger field $L$. And $L$ is the smallest field that works (i.e. p(x) doesn't split into linear factors in any subfield of $L$)

THEN any irreducible polynomial $q(x)$ with coefficients in $F$, which has a linear factor which can be extracted within $L$, will split completely into linear factors in $L$. So $L$ is a splitting field for lots of polynomials - and since the splitting property no longer belongs only to the particular polynomial we started with, $L$ is often called a splitting field without reference to any original polynomial.

Answer 2

A polynomial with coefficients i a field $K$ splits over $L$ if it can be written as a product of linear factors there. Simply put, The polynomial has "all its zeroes" in $L$. The polynomial $X^2-2$ does not split over $\mathbb Q$. But it splits over $\mathbb Q[\sqrt 2]$. The Fundamental Theorem of Algebra can be formulated as: Every polynomial with real coefficiens splits in $\mathbb C$.