What is the difference between "into isomorphism"and "onto isomorphism"?

Question

Kronecker's theorem: Let $F$ be a field and $f(x)$ a nonconstant polynomial in $F[x]$. Then there is an extension field $E$ of $F$ in which $f(x)$ has zero.

Proof: Since $F[x]$ is a unique factorization domain, $f(x)$ has an irreducible factor say $p(x)$. Now consider the ideal $\langle p(x)\rangle$ of the ring of polynomials generated by $p(x)$. Since $\langle p(x)\rangle$ is irreducible over $F$ then the ideal $\langle p(x)\rangle$ is a maximal ideal of $F[x]$. Consequently, $F[x]/\langle p(x)\rangle$ is a field.

Write $E=F[x]/\langle p(x)\rangle$.

Now we shall show that the field $E$ satisfies every part of the theorem. Now consider the map

$\phi:F\rightarrow E$ defined as $\phi (a)=a+\langle p(x)\rangle$.

This mapping is well defined as any element $a\in F$ can be regarded as a constant polynomial in $F[x]$.

$\phi$ is injective

for $\phi (a)=\phi(b)$

$\implies a+\langle p(x)\rangle=b+\langle p(x)\rangle$

$\implies a-b\in\langle p(x)\rangle$

$a-b=f(x)p(x)$ for some $f(x) \in F[x]$.

Since $\deg(p(x))\geq 1$ then $\deg(f(x)g(x))\geq 1$ while $\deg(a-b)=0$. Hence, $f(x)=0$. Consequently, $a-b=0 \implies a=b$.

Clearly, $\phi$ is an homomorphism.

Thus $\phi$ is an isomorphism from $F$ into $E$.

What is the difference between the phrases “$\phi$ is an isomorphism from $F$ into $F'$” and “$\phi$ is an isomorphism from $F$ onto $F'$”?

Actually, I'm proving Kronecker's theorem and the first phrase arose in the middle of the proof (last line). I got confused as it stated the first phrase just by showing $\phi$ is a homomorphism and injective without showing it is surjective.

Please clarify my doubt, if possible with the help of an example.

Answer 1

It could be just a matter of emphasis (but do see the comments below), so it would be more about writing style than mathematics. If $\phi$ is an isomorphism, then it is already both injective and surjective, but I would imagine that the author is thinking something like

• Saying "$\phi$ is an isomorphism from $F$ into $F'$" emphasizes to the reader that the target of the map $\phi$ is $F'.$ Using the term into can be important when emphasizing that a map is well defined. Like you can declare that $F'$ is the codomain of $\phi$ all you want, but when you actually define what map $\phi$ does to elements of $F$, the result better be in $F'$. Sometimes there is a little work to showing that $\phi(x) \in F'$ for all $x \in F$, and saying that $\phi$ maps $F$ into $F'$ captures the idea that this was checked.

• Saying "$\phi$ is an isomorphism from $F$ onto $F′$" emphasizes that the map is surjective, which is possibly the reason the author brought up the fact that $\phi$ is an isomorphism in that part of the argument.

Answer 2

It's impossible to know without having a reference to the textbook or lecture notes you're reading.

Some texts use “isomorphism” where the more common terminology, nowadays, is “injective homomorphism” or “monomorphism”. Apparently, your textbook or lecture notes fall in this category. You should check where the book defines “isomorphism”; most probably it says the equivalent of “injective homomorphism”: this terminology was rather common until a few decades ago.

The first part proves that $\phi$ is an injective homomorphism and the second part proves surjectivity.

Actually, proving $\phi$ is injective is not needed: as soon as $\phi$ is a homomorphism that maps $1$ into $1$, it is automatically injective, because its kernel is an ideal and the ideals in a field $F$ are just $\{0\}$ and $F$.